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M´ ethodes formelles pour les ´ equations aux d´ eriv´ ees partielles

Daniel Robertz

Centre for Mathematical Sciences Plymouth University

JNCF Luminy 2018

Desiderata

Given a system of differential equations, we would like to be able to determine all analytic solutions;

• btain an overview of all consequences of the system;

in particular, given another differential equation, decide whether it is a consequence of the system or not; among the consequences find the ones which involve only certain specified unknowns.

JNCF Luminy 2018

Theorem (Cauchy-Kovalevskaya, 1875)

The Cauchy problem                                              ∂u1 ∂z1 =

n

• j=2

m

• k=1

a1,j,k(z2, . . . , zn, u1, . . . , um) ∂uk ∂zj + b1(z2, . . . , zn, u1, . . . , um), . . . ∂um ∂z1 =

n

• j=2

m

• k=1

am,j,k(z2, . . . , zn, u1, . . . , um) ∂uk ∂zj + bm(z2, . . . , zn, u1, . . . , um), u1(0, z2, . . . , zn) = 0 for all z2, . . . , zn, . . . um(0, z2, . . . , zn) = 0 for all z2, . . . , zn, where ai,j,k and bi are real analytic functions around the origin of Rm+n−1, has a unique real analytic solution (u1, ..., um) in a neighborhood of (z1, ..., zn) = (0, ..., 0).

JNCF Luminy 2018

• C. M´

eray, D´ emonstration g´ en´ erale de l’existence des int´ egrales des ´ equations aux d´ eriv´ ees partielles, Journal de math´ ematiques pures et appliqu´ ees, 3e s´ erie, tome VI:235–265, 1880.

• C. Riquier,

Les syst` emes d’´ equations aux d´ eriv´ ees partielles, Gauthiers-Villars, Paris, 1910.

• M. Janet,

Sur les syst` emes d’´ equations aux d´ eriv´ ees partielles, Gauthiers-Villars, Paris, 1920. Freely available from http://www.numdam.org/item?id=THESE_1920__19__1_0.

• M. Janet,

Le¸ cons sur les syst` emes d’´ equations aux d´ eriv´ ees partielles, Cahiers Scientifiques IV. Gauthiers-Villars, Paris, 1929.

JNCF Luminy 2018

• J. M. Thomas,

Differential Systems, AMS Colloquium Publications, vol. XXI, 1937.

• J. F. Ritt.

Differential Algebra, American Mathematical Society, New York, N. Y., 1950.

• E. R. Kolchin,

Differential algebra and algebraic groups, Academic Press, 1973.

• A. Seidenberg,

An elimination theory for differential algebra,

• Univ. California Publ. Math. (N.S.), 3:31–65, 1956.

JNCF Luminy 2018

Systems of PDEs

A differential system S is given by p1 = 0, p2 = 0, . . . , ps = 0, q1 = 0, q2 = 0, . . . , qt = 0, where p1, ..., ps and q1, ..., qt are polynomials in u1, ..., um of z1, ..., zn and their partial derivatives. Ω open and connected subset of Cn with coordinates z1, . . . , zn The solution set of S on Ω is SolΩ(S) := { f = (f1, . . . , fm) | fk : Ω → C analytic, k = 1, . . . , m, pi(f) = 0, qj(f) = 0, i = 1, . . . , s, j = 1, . . . , t } . Appropriate choice of Ω is possible only after formal treatment.

JNCF Luminy 2018

Systems of linear PDEs

For some l, m, n ∈ N, some ring D

• f differential operators,

some matrix of operators R ∈ Dl×m and some left D-module F we can write R u = 0 , where u =      u1 u2 . . . um      , (1) for the unknown functions ui = ui(z1, . . . , zn) ∈ F, i = 1, . . . , m. Consequences of (1): the left D-linear combinations of the rows of R, i.e., the elements of D1×l R.

JNCF Luminy 2018

Linear PDEs

Example

• f a system of linear PDEs with constant coefficients:

       ∂2u ∂x2 = 0 , ∂2u ∂y2 + ∂u ∂x + ∂u ∂y = 0 , where u = u(x, y) depends on x = z1 and y = z2. Choose D = K[∂x, ∂y], where K ∈ {Q, R, C, . . .} and where ∂x and ∂y are the partial differential operators with respect to x and y, respectively. The multiplication in D is composition of operators.

JNCF Luminy 2018

Linear PDEs

Example

• f a system of linear PDEs with non-constant coefficients:

       ∂3u ∂x ∂y2 − ∂3u ∂y3 − (2y + 1) ∂2u ∂y2 − 4 ∂u ∂y = 0 , ∂3u ∂x2 ∂y − ∂3u ∂y3 − 2 (2y + 1) ∂2u ∂x ∂y + (4y2 + 4y − 5) ∂u ∂y = 0 . Choose K to be Q(x, y) or the field or meromorphic functions on some

• pen and connected subset Ω of C2.

Moreover, we let D = K∂x, ∂y be the ring of differential operators

• i,j≥0

ai,j ∂i

x ∂j y ,

ai,j ∈ K , (skew) polynomials in ∂x and ∂y, with non-comm. composition.

JNCF Luminy 2018

Linear PDEs

Linearizing the system on nonlinear PDEs        ∂u ∂x − u2 = 0 , ∂2u ∂y2 − u3 = 0 , (2) for one unknown function u of x and y, we obtain        ∂U ∂x − 2 u U = 0 , ∂2U ∂y2 − 3 u2 U = 0 , (3) for one unknown function U of x and y, where u is a solution of (2). Preparatory treatment of the nonlinear system (2) is necessary to deal with the linearized system (3).

JNCF Luminy 2018

Linear PDEs

Thomas decomposition

• splitting system (2) into

ux − u2 = 0 { ∂x, ∂y } 2 uy2 − u4 = 0 { ∗ , ∂y } u = 0 u = 0 { ∂x, ∂y } Define D = Q( √ 2)[u, ux, uy, ux,x, ux,y, uy,y, . . .] and the ideal I of D which consists of all D-linear combinations of ux − u2 , ∂x

• ux − u2

, ∂y

• ux − u2

, ∂2

x

• ux − u2

, . . . uy −

√ 2 2 u2 , ∂x

• uy −

√ 2 2 u2

, ∂y

• uy −

√ 2 2 u2

, ∂2

x

• uy −

√ 2 2 u2

, . . . ⇒ D/I integral domain, define K = Quot(D/I), D = K∂x, ∂y.

(Instead of uy −

√ 2 2 u2 one may also choose uy + √ 2 2 u2.) JNCF Luminy 2018

• 1. Janet’s algorithm

JNCF Luminy 2018

Janet’s algorithm for linear PDEs

       ∂2u ∂x∂y − ∂u ∂y = ∂2u ∂x2 − ∂u ∂y = find: u = u(x, y) analytic u(x, y) = a0,0 + a1,0 x + a0,1 y + a2,0 x2

2! + a1,1 xy 1!1! + a0,2 y2 2! + . . .

JNCF Luminy 2018

Janet’s algorithm for linear PDEs

       ∂2u ∂x∂y − ∂u ∂y = ∂2u ∂x2 − ∂u ∂y = find: u = u(x, y) analytic u(x, y) = a0,0 + a1,0 x + a0,1 y + a2,0 x2

2! + a1,1 xy 1!1! + a0,2 y2 2! + . . .

JNCF Luminy 2018

Janet’s algorithm for linear PDEs

       ∂2u ∂x∂y − ∂u ∂y = ∂2u ∂x2 − ∂u ∂y = find: u = u(x, y) analytic

∂ ∂x

• ∂2u

∂x∂y − ∂u ∂y

• − ∂

∂y

• ∂2u

∂x2 − ∂u ∂y

• =

∂2u ∂y2 − ∂u ∂y = 0

u(x, y) = a0,0 + a1,0 x + a0,1 y + a2,0 x2

2! + a1,1 xy 1!1! + a0,2 y2 2! + . . .

JNCF Luminy 2018

Janet’s algorithm for linear PDEs

       ∂2u ∂x∂y − ∂u ∂y = ∂2u ∂x2 − ∂u ∂y = find: u = u(x, y) analytic

∂ ∂x

• ∂2u

∂x∂y − ∂u ∂y

• − ∂

∂y

• ∂2u

∂x2 − ∂u ∂y

• =

∂2u ∂y2 − ∂u ∂y = 0

u(x, y) = a0,0 + a1,0 x + a0,1 y + a2,0 x2

2! + a1,1 xy 1!1! + a0,2 y2 2! + . . .

JNCF Luminy 2018

Janet’s algorithm for linear PDEs

       ∂2u ∂x∂y − ∂u ∂y = ∂2u ∂x2 − ∂u ∂y = find: u = u(x, y) analytic

∂ ∂x

• ∂2u

∂x∂y − ∂u ∂y

• − ∂

∂y

• ∂2u

∂x2 − ∂u ∂y

• =

∂2u ∂y2 − ∂u ∂y = 0

u(x, y) = a0,0 + a1,0 x + a0,1 y + a2,0 x2

2! + a1,1 xy 1!1! + a0,2 y2 2! + . . .

Janet’s algorithm computes a vector space basis for power series solutions

(Maurice Janet, ∼ 1920)

JNCF Luminy 2018

Janet’s algorithm for linear PDEs

   uy,y = 0 ux,x − yuz,z = 0

A B

is equivalent to                                        uy,y = 0 ux,x − yuz,z = 0 uy,z,z = 0 ux,y,y = 0 uz,z,z,z = 0 ux,y,z,z = 0 ux,z,z,z,z = 0

A B

1 2 (∂2 x − y∂2 z)A − 1 2 ∂2 yB

∂xA

1 2 (∂4 x − 2y∂2 x∂2 z + y2∂4 z)A − 1 2 (∂2 x∂2 y − y∂2 y∂2 z + 2∂y∂2 z)B 1 2 (∂3 x − y∂x∂2 z)A − 1 2 ∂x∂2 yB 1 2 (∂5 x − 2y∂3 x∂2 z + y2∂x∂4 z)A − 1 2 (∂3 x∂2 y + y∂x∂2 y∂2 z − 2∂x∂y∂2 z)B

Taylor coeff’s for 1, z, y, x, z2, yz, xz, xy, z3, xz2, xyz, xz3 arbitrary, all other coeff’s determined by linear equations

JNCF Luminy 2018

Strategy

D = K[∂1, . . . , ∂n], K field I ideal of D (or, more generally: M submodule of Dm) goal: compute “good” generating set for I (Janet basis) sort terms in a polynomial in a way compatible with multiplication total ordering >

• n

M := Mon(∂1, . . . , ∂n) := {∂i | i ∈ (Z≥0)n} no infinitely descending chains of monomials use highest terms lm(p), lm(q) to decide “divisibility”

JNCF Luminy 2018

Multiple closed sets of monomials

M := Mon(∂1, . . . , ∂n) := {∂i | i ∈ (Z≥0)n} S ⊆ M is M-multiple closed if m s ∈ S ∀ m ∈ M, s ∈ S M-multiple closed set generated by ∂1∂2

2,

∂3

1∂2,

∂4

1

=: ∂1∂2

2, ∂3 1∂2, ∂4 1 M

JNCF Luminy 2018

Multiple closed sets of monomials

Lemma

Every M-multiple closed set S ⊆ M has a finite generating set. Proof. Every seq. F : m1, m2, m3 . . . ∈ M s.t. mi | mj ∀ i < j is finite. Induction: n = 1: clear. n − 1 → n: Let m1 = ∂a1

1 · · · ∂an n .

Define subsequence F (j,d) : mi = ∂b1

1 · · · ∂d j · · · ∂bn n

We have:

• 1≤j≤n
• 0≤d≤aj

{F (j,d)} = {F} By induction, the {F (j,d)} are finite.

JNCF Luminy 2018

Multiple closed sets of monomials

Lemma

Every M-multiple closed set S ⊆ M has a finite generating set.

Cor.

Every ascending sequence of M-multiple closed sets becomes stationary. Given a finite generating set {p1, . . . , pr} for I K[∂1, . . . , ∂n], Janet’s algorithm computes S0 ⊆ S1 ⊆ . . . ⊆ Sk = lm(I) (all M-multiple closed) where S0 is generated by lm(p1), . . . , lm(pr) ⇒ termination

JNCF Luminy 2018

Decomposition into disjoint cones

Def.

Let C ⊆ Mon(∂1, . . . , ∂n), µ ⊆ {∂1, . . . , ∂n}. (C, µ) is a cone if ∃ v ∈ C s.t. C = Mon(µ) v Variables in µ: multiplicative variables (for v) Variables in {∂1, . . . , ∂n} − µ: non-multiplicative variables (for v) Dimension of (C, µ): |µ|

JNCF Luminy 2018

Decomposition into disjoint cones

Def.

Let S ⊆ M = Mon(∂1, . . . , ∂n). { (C1, µ1), . . . , (Cr, µr) } ⊂ P(M) × P({∂1, . . . , ∂n}) is a decomposition of S into disjoint cones if each (Ci, µi) is a cone and S = ˙ r

i=1Ci.

{ (v1, µ1), . . . , (vr, µr) } is a

• decomp. of S into disj. cones

if { (Mon(µ1) v1, µ1), . . . , (Mon(µr) vr, µr) } is one.

JNCF Luminy 2018

Decomposition into disjoint cones

Strategy of Janet’s algorithm: Decompose M-multiple closed sets S into disjoint cones. S = ∂1∂2

2, ∂3 1∂2, ∂4 1 M

decomposition: ∂1∂2

2

{ ∗ , ∂2 } ∂2

1∂2 2

{ ∗ , ∂2 } ∂3

1∂2

{ ∗ , ∂2 } ∂4

1

{ ∂1, ∂2 } This can also be done for Mon(∂1, . . . , ∂n) − S.

JNCF Luminy 2018

Janet division

The possible ways of decomposing M-multiple closed sets into disjoint cones are studied as involutive divisions (Gerdt, Blinkov et. al.) Janet division: Let G ⊂ M = Mon(∂1, . . . , ∂n) be finite. For a cone with vertex v = ∂a1

1 · · · ∂an n

∈ G ∂i is a multiplicative variable iff ai = max{ bi | ∂b ∈ G; bj = aj ∀ j < i }.

JNCF Luminy 2018

Janet division

For v = ∂a1

1 · · · ∂an n :

∂i ∈ µ ⇐ ⇒ ai = max{ bi | ∂b ∈ G; bj = aj ∀ j < i }. Example: G = { ∂2 ∂3, ∂1 ∂2 ∂3, ∂2

1 ∂2 ∂3, ∂2 1 ∂2 2 }

∂2 ∂3 ∂1 ∂2 ∂3 ∂2

1 ∂2 ∂3

∂2

1 ∂2 2

JNCF Luminy 2018

Janet division

For v = ∂a1

1 · · · ∂an n :

∂i ∈ µ ⇐ ⇒ ai = max{ bi | ∂b ∈ G; bj = aj ∀ j < i }. Example: G = { ∂2 ∂3, ∂1 ∂2 ∂3, ∂2

1 ∂2 ∂3, ∂2 1 ∂2 2 }

∂2 ∂3 ∗ ∂2 ∂3 ∂1 ∂2 ∂3 ∗ ∂2 ∂3 ∂2

1 ∂2 ∂3

∂1 ∗ ∂3 ∂2

1 ∂2 2

∂1 ∂2 ∂3

JNCF Luminy 2018

Decomposition into disjoint cones

Decompose(G, η) G ⊂ Mon(∂1, . . . , ∂n), ∅ = η ⊆ {∂1, . . . , ∂n}

G ← {g ∈ G |∃ h ∈ G : h | g} if |G| ≤ 1

• r

|µ| = 1 then return {(m, η) | m ∈ G} else y ← ya with a = min{i | 1 ≤ i ≤ n, yi ∈ η} d ← max{degy(g) | g ∈ G} Gi ← {g ∈ G | degy(g) = i}, i = 0, . . . , d Gi ← Gi ∪ i−1

j=0 {yi−jg | g ∈ Gj},

i = 1, . . . , d Td ← { (m, ζ ∪ {y}) | (m, ζ) ∈ Decompose(Gd, η − {y}) } Ti ← Decompose(Gi, η − {y}), i = 0, . . . , d − 1 return d

i=0 Ti

fi

JNCF Luminy 2018

Janet reduction

NF(p, T, >) p ∈ K[∂1, . . . , ∂n], T = { (d1, µ1), . . . , (ds, µs) }

r ← 0 while p = 0 do if ∃ (d, µ) ∈ T : lm(p) ∈ Mon(µ) lm(d) then p ← p − lc(p)

lc(d) lm(p) lm(d) d

else r ← r + lc(p) lm(p) p ← p − lc(p) lm(p) fi

• d

return r Disjoint cones ⇒ course of algorithm is uniquely determined

JNCF Luminy 2018

Janet’s algorithm

JanetBasis(F, >) F ⊆ K[∂1, . . . , ∂n] finite

G ← F do G ← auto-reduce G J ← { (p1, µ1), . . . , (pr, µr) } s.t. { (lm(p1), µ1), . . . , (lm(pr), µr) } Janet decomposition of lm(G) M P ← { NF(∂ · p, J) | (p, µ) ∈ J, ∂ ∈ µ } (passivity check) G ← { p | (p, µ) ∈ J } ∪ P while P = {0} return J

JNCF Luminy 2018

Janet basis

Janet basis J = { (p1, µ1), . . . , (pr, µr) } for I = F Invariant of the loop: G (or {p1, . . . , pr}) always forms a gen. set for I

r

• i=1

Mon(µi)pi is a K-basis of I. Linear independence: clear. p ∈ I: p =

r

• i=1

ci pi

JNCF Luminy 2018

Example

Let I := g1, g2 K[x, y], g1 := x2 − y, g2 := xy − y. Let > be degrevlex, x > y. Decomposition into disjoint cones of lm(g1), lm(g2) : { (x2, {x, y}), (xy, {y}) } f := x · g2 = x2y − xy ∈ I, f =

2

• i=1

ci gi? Reduction of f modulo g1, g2 yields: g3 := y2 − y ∈ I { (g1, {x, y}), (g2, {y}), (g3, {y}) } (minimal) Janet basis for I

JNCF Luminy 2018

Linear PDEs

Linearizing the system on nonlinear PDEs        ∂u ∂x − u2 = 0 , ∂2u ∂y2 − u3 = 0 , (4) for one unknown function u of x and y, we obtain        ∂U ∂x − 2 u U = 0 , ∂2U ∂y2 − 3 u2 U = 0 , (5) for one unknown function U of x and y, where u is a solution of (4). Preparatory treatment of the nonlinear system (4) is necessary to deal with the linearized system (5).

JNCF Luminy 2018

Linear PDEs

Thomas decomposition

• splitting system (4) into

ux − u2 = 0 { ∂x, ∂y } 2 uy2 − u4 = 0 { ∗ , ∂y } u = 0 u = 0 { ∂x, ∂y } Define D = Q( √ 2)[u, ux, uy, ux,x, ux,y, uy,y, . . .] and the ideal I of D which consists of all D-linear combinations of ux − u2 , ∂x

• ux − u2

, ∂y

• ux − u2

, ∂2

x

• ux − u2

, . . . uy −

√ 2 2 u2 , ∂x

• uy −

√ 2 2 u2

, ∂y

• uy −

√ 2 2 u2

, ∂2

x

• uy −

√ 2 2 u2

, . . . ⇒ D/I integral domain, define K = Quot(D/I), D = K∂x, ∂y.

(Instead of uy −

√ 2 2 u2 one may also choose uy + √ 2 2 u2.) JNCF Luminy 2018

Example

∂ ∂x ∂2U ∂y2 − 3 u2 U

• =

∂3U ∂x ∂y2 − 3

• 2 u ∂u

∂x U + u2 ∂U ∂x

• =

∂3U ∂x ∂y2 − 6 u3 U − 6 u3 U . and ∂2 ∂y2 ∂U ∂x − 2 u U

• =

∂3U ∂x ∂y2 − 2 ∂2u ∂y2 U + 2 ∂u ∂y ∂U ∂y + u ∂2U ∂y2

• =

∂3U ∂x ∂y2 − 2 u3 U − 2 √ 2 u2 ∂U ∂y − 6 u3 U Hence, we obtain ∂ ∂x ∂2U ∂y2 − 3 u2 U

• − ∂2

∂y2 ∂U ∂x − 2 u U

• = 2

√ 2 u2 ∂U ∂y − 4 u3 U , which yields the consequence ∂U ∂y − √ 2 u U = 0 .

JNCF Luminy 2018

Example

Janet basis: Ux − 2 u U = 0, { ∂x, ∂y }, Uy ∓ √ 2 u U = 0, { ∗ , ∂y }. Substituting u(x, y) = 2 −2x ± √ 2y + c, c ∈ R, for u in this Janet basis results in a system of linear PDEs for U whose analytic solutions are given by U(x, y) = C (−2x ± √ 2y + c)2 , C ∈ R.

JNCF Luminy 2018

(Generalized) Hilbert series

Janet basis J = { (p1, µ1), . . . , (pr, µr) } for I We have lm(I) = lm(p1), . . . , lm(pr) M. Generalized Hilbert series HI(∂1, . . . , ∂n) =

r

• i=1

lm(pi)

• ∂j∈µi

1 1 − ∂j enumerates a K-basis of lm(I) . HI(t, . . . , t) is the usual Hilbert series.

JNCF Luminy 2018

Example

Janet basis for I J = { (∂1∂2

2, {∂2}), (∂2 1∂2 2, {∂2}), (∂3 1∂2, {∂2}), (∂4 1, {∂1, ∂2}) }

generalized Hilbert series: ∂1∂2

2

1 − ∂2 + ∂2

1∂2 2

1 − ∂2 + ∂3

1∂2

1 − ∂2 + ∂4

1

(1 − ∂1)(1 − ∂2)

JNCF Luminy 2018

Hilbert polynomial

Janet basis of M: { (p1, µ1), . . . , (pr, µr) } HM(t, . . . , t) =

• k≥0

dimK Mk tk =

r

• i=1

tdeg(pi) 1 (1 − t)|µi| =

r

• i=1

tdeg(pi)

j≥0

|µi| + j − 1 j

• tj
• Coeff. of tk in HM(t, . . . , t) ?

For k ≥ max{deg(pi) | i = 1, . . . , r}: dimK Mk =

r

• i=1

|µi| + k − deg(pi) − 1 k − deg(pi)

• JNCF Luminy 2018

Example

S = lm(I) Decomposition of Mon(∂1, . . . , ∂n) − S into disjoint cones generalized Hilbert series enum. a K-basis of K[∂1, . . . , ∂n]/I generalized Hilbert series: 1 1 − ∂2 + ∂1 + ∂1∂2 + ∂2

1 + ∂2 1∂2 + ∂3 1

Hilbert polynomial:

6

• i=1

|µi| + k − deg(pi) − 1 k − deg(pi)

• = 1

JNCF Luminy 2018

Power series solutions

∂2u ∂x ∂y = 0, { ∗ , ∂y , ∂z }, ∂3u ∂x2 ∂y = 0, { ∗ , ∂y , ∂z }, ∂4u ∂x3 ∂z = 0, { ∂x , ∗ , ∂z }, ∂4u ∂x3 ∂y = 0, { ∂x , ∂y , ∂z }. Janet decomposition of the set of parametric derivatives / generalized Hilbert series: 1, { ∗ , ∂y , ∂z }, ∂x, { ∗ , ∗ , ∂z }, ∂2

x,

{ ∗ , ∗ , ∂z }, ∂3

x,

{ ∂x , ∗ , ∗ }.

1 (1−∂y)(1−∂z) + ∂x 1−∂z + ∂2

x

1−∂z + ∂3

x

1−∂x .

Accordingly, a formal power series solution u is uniquely determined as u(x, y, z) = f0(y, z) + x f1(z) + x2 f2(z) + x3 f3(x) by any choice of formal power series f0, f1, f2, f3 of the indicated variables.

JNCF Luminy 2018

Power series solutions

         ∂2u ∂x2 = f(x, y) ∂2u ∂y2 = g(x, y) complete:                  ∂2u ∂x2 = f(x, y) { ∂x , ∂y }, ∂2u ∂y2 = g(x, y) { ∗ , ∂y }, ∂3u ∂x∂y2 = ∂g ∂x { ∗ , ∂y } integrability condition: ∂2 ∂x2 ∂2u ∂y2

• = ∂2g

∂x2 ∂2 ∂y2 ∂2u ∂x2

• = ∂2f

∂y2          = ⇒ ∂2f ∂y2 = ∂2g ∂x2 .

JNCF Luminy 2018

Power series solutions

Janet decomposition of the set of parametric derivatives: 1, { ∗ , ∗ }, ∂x, { ∗ , ∗ }, ∂y, { ∗ , ∗ }, ∂x∂y, { ∗ , ∗ }. initial conditions:                        u(0, 0) = a0,0, ∂u ∂x (0, 0) = a1,0, ∂u ∂y (0, 0) = a0,1, ∂2u ∂x∂y (0, 0) = a1,1 u(x, y) = a0,0+a0,1 y+a1,0 x+a1,1 x y+

• i≥2, j≥0

ai,j xi i! yj j! +

• j≥2

a0,j yj j! +

• j≥2

a1,j x yj j!

JNCF Luminy 2018

Power series solutions

Janet decomposition of the set of parametric derivatives: 1, { ∗ , ∗ }, ∂x, { ∗ , ∗ }, ∂y, { ∗ , ∗ }, ∂x∂y, { ∗ , ∗ }. initial conditions:                        u(0, 0) = a0,0, ∂u ∂x (0, 0) = a1,0, ∂u ∂y (0, 0) = a0,1, ∂2u ∂x∂y (0, 0) = a1,1 u(x, y) = a0,0+...+a1,1 x y+ x x f(x, y) dx dx+ y y g(0, y) dy dy+x y y ∂g ∂x(0, y) dy dy

JNCF Luminy 2018

Desiderata

Given a system of differential equations, we would like to be able to determine all analytic solutions;

• btain an overview of all consequences of the system;

in particular, given another differential equation, decide whether it is a consequence of the system or not; among the consequences find the ones which involve only certain specified unknowns.

JNCF Luminy 2018

Elimination

Lemma

J ⊆ R := K[X1, . . . , Xn, Y1, . . . , Ym] Janet basis w.r.t. any term order. For any 0 = p ∈ R let lm(p) be its leading monomial. If { p ∈ J | p ∈ K[Y1, . . . , Ym] } = { p ∈ J | lm(p) ∈ K[Y1, . . . , Ym] }, then J ∩ K[Y1, . . . , Ym] generates J ∩ K[Y1, . . . , Ym]. Proof. Let 0 = p ∈ J ∩ K[Y1, . . . , Ym]. Since J is a Janet basis, ∃ q ∈ J, lm(q) ∈ K[Y1, . . . , Ym], lm(q) | lm(p). By assumption, q ∈ K[Y1, . . . , Ym]. Reduction p → 0 in K[Y1, . . . , Ym].

• JNCF Luminy 2018

Free resolution

Janet basis J = { (p1, µ1), . . . , (pr, µr) } for I We have ∂j pi =

k αi,j,k pk,

∂j ∈ µi, αi,j,k ∈ K[µk] Define π : D|J| → D : ˆ pi → pi. (ˆ pi std. gen.)

Prop.

∂j ˆ pi −

• k

αi,j,k ˆ pk, ∂j ∈ µi, i = 1, . . . , r, form a Janet basis of ker π for a suitable monomial ordering. construction of a free resolution of D/I.

JNCF Luminy 2018

Example

D = K[∂1, ∂2, ∂3], ∂1 > ∂2 > ∂3, I = (∂1, ∂2, ∂3) Janet basis: ∂1 ∂1 ∂2 ∂3 ∂2 ∗ ∂2 ∂3 ∂3 ∗ ∗ ∂3 normal form computation: ∂1 · ∂2 − ∂2 · ∂1 ∂1 · ∂3 − ∂3 · ∂1 ∂2 · ∂3 − ∂3 · ∂2 D1×3

  

−∂2 ∂1 −∂3 ∂1 −∂3 ∂2

  

− − − − − − − − − − − − − − − → D1×3

  

∂1 ∂2 ∂3

  

− − − − − − → D − → D/I − → 0

JNCF Luminy 2018

Example

D = K[∂1, ∂2, ∂3], ∂1 > ∂2 > ∂3, I = (∂1, ∂2, ∂3) Janet basis: [−∂2 ∂1 0 ] ∂1 ∂2 ∂3 [−∂3 ∂1 ] ∂1 ∂2 ∂3 [ −∂3 ∂2 ] ∗ ∂2 ∂3 normal form computation: ∂1 · [ 0 −∂3 ∂2] − ∂2 · [−∂3 ∂1] + ∂3 · [−∂2 ∂1 0] 0 → D ( ∂3

−∂2 ∂1 )

− − − − − − − − − − − → D1×3

  

−∂2 ∂1 −∂3 ∂1 −∂3 ∂2

  

− − − − − − − − − − − − − − → D1×3

  

∂1 ∂2 ∂3

  

− − − − − → D → D/I → 0

JNCF Luminy 2018

Free resolution

Prop.

∂j ˆ pi −

• k

αi,j,k ˆ pk, ∂j ∈ µi, i = 1, . . . , r, form a Janet basis of ker π w.r.t. ≺. Choose a total order ≪ on J s.t. pk ≪ pl if ∃ path from pl to pk in the Janet graph. ∂1 ∂2 ∂3 ∂1 ∂2 ∂1 ∂i ˆ pk ≺ ∂j ˆ pl :⇐ ⇒

• ∂i lm(pk) < ∂j lm(pl)
• r

∂i lm(pk) = ∂j lm(pl) and pk ≪ pl

JNCF Luminy 2018

Consequences

{ (p1, µ1), . . . , (pr, µr) } Janet basis for I R can decide ideal membership normal form for residue classes modulo I enumeration of a K-basis of I and a K-basis of R/I (generalized Hilbert series) can easily determine Hilbert polynomial can read off a free resolution of R/I every Janet basis is a Gr¨

• bner basis

JNCF Luminy 2018

Janet bases over Z

NF(p, T, ≺) p ∈ Z[x1, . . . , xn], T = { (d1, µ1), . . . , (dl, µl) }

r ← 0 while p = 0 do if ∃ (d, µ) ∈ T : lm(p) ∈ Mon(µ) d then write lc(p) = a · lc(d) + b, |b| < | lc(d)| if a = 0 then p ← p − a lm(p)

lm(d) d

else move leading term from p to r fi else move leading term from p to r fi

• d

return r

JNCF Luminy 2018

Janet bases for Ore algebras

Skew polynomial ring A[∂; σ, δ]: A domain and K-algebra σ : A → A K-algebra endomorphism δ : A → A σ-derivation, i.e. δ(a b) = σ(a) δ(b) + δ(a) b, a, b ∈ A A[∂; σ, δ] =

• fin.

ai ∂i | ai ∈ A, i ∈ Z≥0

• with commutation rule

∂ a = σ(a) ∂ + δ(a), a ∈ A

JNCF Luminy 2018

Janet bases for Ore algebras

Ore algebra D = A[∂1; σ1, δ1] . . . [∂m; σm, δm]: A = K

• r

A = K[x1, . . . , xn] σi : D → D K-algebra endomorphisms δi : D → D σi-derivations A[∂1; σ1, δ1] . . . [∂m; σm, δm] =

• fin.

ai ∂i | ai ∈ A, i ∈ (Z≥0)m

• with commutation rules

∂i a = σi(a) ∂i + δi(a), a ∈ A, ∂i ∂j = ∂j ∂i

JNCF Luminy 2018

Janet bases for Ore algebras

Weyl algebra:

• rdinary differential equations

A1 = K[t][ d

dt] d dt a = a d dt + da dt

Weyl algebra: partial differential equations An = K[x1, . . . , xn][∂1, . . . , ∂n] ∂i xj = xj ∂i + δij Bn = K(x1, . . . , xn)[∂1, . . . , ∂n] Shift operators: difference equations Sh = K[t][δh] δh t = (t − h) δh combinations . . .

JNCF Luminy 2018

Janet bases for Ore algebras

D = K[x1, . . . , xn][∂1, . . . , ∂m] I left ideal of D generated by p1, . . . , pr normal form for elements of D: use ∂i xj = σi(xj) ∂i + . . . to move all ∂i to the right of every xj M := { xi ∂j | i ∈ (Z≥0)n, j ∈ (Z≥0)m } consider M-multiple closed set generated by the normal forms of lm(pi), i = 1, . . . , r

• decomp. into disj. cones as before

reduction: all multiplications from the left

JNCF Luminy 2018

Janet bases for Ore algebras

D = K[x1, . . . , xn][∂1, . . . , ∂m] I left ideal of D generated by p1, . . . , pr For termination of the algorithm, assume that ∂i xj = (ci,j xj + di,j) ∂i + ei,j where ci,j ∈ K − {0}, di,j ∈ K, ei,j ∈ K[x1, . . . , xn] with deg(ei,j) ≤ 1

JNCF Luminy 2018

Involutive

Janet (-like Gr¨

• bner) bases for submodules of free modules over a

commutative polynomial ring coefficients: rationals or finite fields and field extensions, and rational integers Janet division, Janet-like division term orderings: degrevlex, plex TOP / POT block / elimination orderings web: http://wwwb.math.rwth-aachen.de/Janet

JNCF Luminy 2018

Involutive

Analogues of Buchberger’s criteria can be selected Interface to C++: call fast routines when needed or switch to fast routines for the whole Maple session Syzygies, Hilbert series, etc. Applications: commutative algebra solving systems of algebraic equations web: http://wwwb.math.rwth-aachen.de/Janet

JNCF Luminy 2018

Main procedures of Involutive

InvolutiveBasis compute Janet(-like Gr¨

• bner) basis

PolInvReduce involutive reduction modulo Janet basis FactorModuleBasis vector space basis of residue class module Syzygies syzygy module PolResolution free resolution PolHilbertSeries, PolHilbertPolynomial, etc. combinatorial devices PolMinPoly, PolRepres, etc. computing in residue class rings

JNCF Luminy 2018

Janet

Janet (-like Gr¨

• bner) bases for linear systems of partial differential

equations Janet division, Janet-like division analogues of Buchberger’s criteria can be selected computational tools for differential operators elementary divisor algorithm for K(x)[∂] (Jacobson normal form) parametric derivatives formal power series solutions, polynomial solutions web: http://wwwb.math.rwth-aachen.de/Janet

JNCF Luminy 2018

Main procedures of Janet

JanetBasis compute Janet(-like Gr¨

• bner) basis

InvReduce involutive reduction modulo Janet basis ParamDeriv parametric derivatives CompCond, Resolution compatibility conditions (syzygies) HilbertSeries, HilbertPolynomial, etc. combinatorial devices SolSeries, PolySol formal power series / polynomial solutions ElementaryDivisors Jacobson normal form

JNCF Luminy 2018

ginv

C++ module for Python

• comp. of Gr¨
• bner bases using involutive algorithms

polynomials, differential / difference equations

• pen source software
• riginated by V. P. Gerdt, Y. A. Blinkov

contributions by LBfM coefficients: rationals or finite fields and some algebraic and transcendental field extensions term orderings: degrevlex (TOP / POT), lex, product orderings see web page for timings web: http://invo.jinr.ru

JNCF Luminy 2018

ginv

import ginv st = ginv.SystemType("Polynomial") im = ginv.MonomInterface("DegRevLex", st, [’x’, ’y’]) ic = ginv.CoeffInterface("GmpZ", st) ip = ginv.PolyInterface("PolyList", st, im, ic) iw = ginv.WrapInterface("CritPartially", ip) iD = ginv.DivisionInterface("Janet", iw) eqs = ["x^2+y^2", ...] basis = ginv.basisBuild("TQ", iD, eqs)

JNCF Luminy 2018

Involutive Basis Algorithm (Gerdt)

choose f ∈ F with the lowest lm(f) w.r.t. ≺ G ← {f}; Q ← F − G do h ← 0 while Q = ∅ and h = 0 do choose p ∈ Q with the lowest lm(p) w.r.t. ≺ Q ← Q − {p}; h ← NF(p, G, ≺) if h = 0 then for all {g ∈ G | lm(g) = xi lm(h), |i| > 0} do Q ← Q ∪ {g}; G ← G − {g} G ← G ∪ {h} Q ← Q ∪ { x · g | g ∈ G, x non-mult. for g } while Q = ∅ return G

JNCF Luminy 2018

Janet-like Gr¨

• bner Bases

Idea: do not store all the prolongations Janet-like division Note: In general, the minimal Gr¨

• bner basis is still a proper subset of the

Janet-like Gr¨

• bner basis.
• V. P. Gerdt, Y. A. Blinkov, Janet-like Monomial Division. Janet-like Gr¨
• bner Bases.

CASC 2005, LNCS 3781, Springer, 2005

JNCF Luminy 2018

Formal Power Series

Let D = K[z1, ..., zn][∂1, ..., ∂l] s.t. Janet bases can be computed.

Theorem

F := homK(D, K) is an injective left D-module. The pairing ( , ) : D × F → K : (d, λ) → λ(d) is non-degenerate: λ ∈ F is uniquely determined by λ(d), d ∈ D λ ∈ F is uniquely determined by λ(m), m ∈ Mon(D)

JNCF Luminy 2018

Formal Power Series

Thm. F := homK(D, K) is an injective left D-module. Proof. By Baer’s criterion, consider w.l.o.g. left ideal I

• f

D and ϕ : I → F. Extension

• ϕ : D → F ?

System of D-linear equations d · λ = ϕ(d), d ∈ I, λ ∈ F. J Janet basis for I ⇒

r

• i=1

Mon(µi)pi is a K-basis of I choose values (m, λ) = λ(m) for m ∈ Mon(D) − lm(I) ⇒ values (lm(p), λ) uniquely determined ⇒ solution λ ∈ F

• ϕ

defined by

• ϕ(1) := λ.

JNCF Luminy 2018

References

• M. Janet,

Le¸ cons sur les syst` emes d’´ equations aux d´ eriv´ ees partielles, Gauthiers-Villars, Paris, 1929

• C. M´

eray, D´ emonstration g´ en´ erale de l’existence des int´ egrales des ´ equations aux d´ eriv´ ees partielles,

• J. de math´

ematiques pures et appliqu´ ees, 3e s´ erie, tome VI, 1880

• C. Riquier,

Les syst` emes d’´ equations aux d´ eriv´ ees partielles, Gauthiers-Villars, Paris, 1910

• J. F. Ritt,

Differential Algebra, Dover, 1966

JNCF Luminy 2018

References

• W. Plesken, D. Robertz,

Janet’s approach to presentations and resolutions for polynomials and linear pdes, Archiv der Mathematik, 84 (1), 2005, pp. 22–37

• Y. A. Blinkov, C. F. Cid, V. P. Gerdt, W. Plesken, D. Robertz,

The Maple Package ”Janet”: I. Polynomial Systems and

• II. Linear Partial Differential Equations,

Proceedings of CASC 2003, pp. 31–40 resp. pp. 41–54 F.-O. Schreyer, Die Berechnung von Syzygien mit dem verallgemeinerten Weierstraßschen Divisionssatz und eine Anwendung auf analytische Cohen-Macaulay-Stellenalgebren minimaler Multiplizit¨ at, Diploma Thesis, Univ. Hamburg, Germany, 1980

JNCF Luminy 2018

References

• D. Robertz,

Formal Computational Methods for Control Theory, PhD thesis, RWTH Aachen University, 2006, available at http://darwin.bth.rwth-aachen.de/opus/volltexte/2006/1586

• D. Robertz,

Janet bases and applications, in: M. Rosenkranz, D. Wang, Gr¨

• bner Bases in Symbolic Analysis,

Radon Series Comp. Appl. Math., de Gruyter, 2007

• D. Robertz,

Noether normalization guided by monomial cone decompositions,

• J. of Symbolic Computation, 44 (10), 2009, pp. 1359–1373
• D. Robertz,

Formal Algorithmic Elimination for PDEs, Habilitationsschrift, accepted by the Faculty of Mathematics, Computer Science and Natural Sciences, RWTH Aachen University, 2012

JNCF Luminy 2018

References

• W. Plesken, D. Robertz,

Constructing Invariants for Finite Groups, Experimental Mathematics, 14 (2), 2005, pp. 175–188

• W. Plesken, D. Robertz,

Representations, commutative algebra, and Hurwitz groups,

• J. Algebra, 300 (2006), 2006, pp. 223–247
• W. Plesken, D. Robertz,

Elimination for coefficients of special characteristic polynomials, Experimental Mathematics 17 (4), 2008, pp. 499–510

• W. Plesken, D. Robertz,

Linear Differential Elimination for Analytic Functions, Mathematics in Computer Science, 4 (2–3), 2010, pp. 231–242

JNCF Luminy 2018

References

• V. P. Gerdt, Y. A. Blinkov,

Involutive bases of polynomial ideals. Minimal involutive bases, Mathematics and Computers in Simulation, 45, 1998

• Y. A. Blinkov, V. P. Gerdt, D. A. Yanovich,

Construction of Janet Bases, I. Monomial Bases, II. Polynomial Bases, Proceedings of CASC 2001

• V. P. Gerdt,

Involutive Algorithms for Computing Gr¨

• bner Bases,
• Proc. “Computational commutative and non-commutative algebraic

geometry” (Chishinau, June 6-11, 2004), IOS Press, 2005

• V. P. Gerdt, Y. A. Blinkov, V. V. Mozzhilkin,

Gr¨

• bner Bases and Generation of Difference Schemes for Partial

Differential Equations, Symmetry, Integrability and Geometry: Methods and Applications, 2006

JNCF Luminy 2018

References

• V. P. Gerdt, D. Robertz,

A Maple Package for Computing Gr¨

• bner Bases for Linear Recurrence

Relations, Nuclear Instruments and Methods in Physics Research A, 559 (1), 2006,

• pp. 215–219
• V. P. Gerdt, D. Robertz,

Consistency of Finite Difference Approximations for Linear PDE Systems and its Algorithmic Verification, in: S. M. Watt (ed.), Proceedings of ISSAC 2010, TU M¨ unchen, Germany, pp. 53–59

• V. P. Gerdt, D. Robertz,

Computation of Difference Gr¨

• bner Bases,

Computer Science Journal of Moldova, 20 (2), 2012, pp. 203–226

JNCF Luminy 2018

References

• F. Chyzak, B. Salvy,

Non-commutative elimination in Ore algebras proves multivariate identities,

• J. Symbolic Computation, 26, 1998
• V. Levandovskyy,

Non-commutative Computer Algebra for polynomial algebras: Gr¨

• bner

bases, applications and implementation, PhD thesis, Univ. Kaiserslautern, Germany, 2005

• V. P. Gerdt, D. A. Yanovich,

Experimental Analysis of Involutive Criteria, “Algorithmic Algebra and Logic 2005”, April 3-6, 2005, Passau, Germany

• J. Apel, R. Hemmecke,

Detecting unnecessary reductions in an involutive basis computation,

• J. Symbolic Computation, 40, 2005

JNCF Luminy 2018

References

• W. W. Adams, P. Loustaunau,

An Introduction to Gr¨

• bner Bases,

AMS, 1994

• T. Becker and V. Weispfenning,

Gr¨

• bner Bases. A Computational Approach to Commutative Algebra,

Springer, 1993

• D. Cox, J. Little, D. O’Shea,

Ideals, Varieties, and Algorithms, Springer, 1992

• D. Eisenbud,

Commutative Algebra with a View Toward Algebraic Geometry, Springer, 1995

JNCF Luminy 2018

References

• B. Malgrange,

Syst` emes ` a coefficients constants, S´ eminaire Bourbaki 246:79–89, 1962–63.

• U. Oberst,

Multidimensional constant linear systems, Acta Appl. Math. 20:1–175, 1990. J.-F. Pommaret and A. Quadrat, Algebraic analysis of linear multidimensional control systems, IMA Journal of Control and Information 16 (3):275–297, 1999. J.-F. Pommaret and A. Quadrat, A functorial approach to the behavior of multidimensional control systems, Applied Mathematics and Computer Science, 13:7–13, 2003. J.-F. Pommaret Partial Differential Control Theory Kluwer, 2001

JNCF Luminy 2018

References

• M. Barakat and D. Robertz,

homalg: A meta-package for homologial algebra, Journal of Algebra and Its Applications 7 (3):299–317, 2008.

• F. Chyzak and A. Quadrat and D. Robertz,

OreModules: A symbolic package for the study of multidimensional linear systems, in: Chiasson, J. and Loiseau, J.-J. (eds.), Applications

• f Time-Delay Systems,

LNCIS 352, 233–264, Springer, 2007.

• T. Cluzeau and A. Quadrat,

OreMorphisms: A homological algebra package for factoring and decomposing linear functional systems, in: Loiseau, J.-J., Michiels, W., Niculescu, S.-I., Sipahi, R. (eds.), Topics in Time-Delay Systems: Analysis, Algorithms and Control, LNCIS, Springer, 2008.

JNCF Luminy 2018

Algebraic Geometry

R → R2 t →

• 2t

t2+1, t2−1 t2+1

• x

y x2 + y2 − 1 = 0 Eliminate t in x = 2t t2 + 1, y = t2 − 1 t2 + 1 . . .

JNCF Luminy 2018

Special Solutions

∂v ∂t + v · ∇v − ν∆v + 1 ρ∇p

= (Navier-Stokes)

∂ρ ∂t + ∇ · (ρv)

= cylindrical coordinates r, θ, z, ρ ≡ 1 (incompressible flow) Ansatz: vi(r, θ, z) = fi(r)gi(θ)hi(z), i = 1, 2, 3 PDE: uux,y − uxuy = 0,

u ∈ {v1, v2, v3}, (x, y) ∈ {(r, θ), (r, z), (θ, z)}

• ne of the many simple systems of the Thomas decomposition:

v(t, r, θ, z) =

• − (t+c2)F1(t)

r

−

r 2(t+c2), (θ+c1)r t+c2

, 0

• ,

p(t, r, θ, z) = (t + c2) ln(r) ˙ F1(t) − (t+c2)2F1(t)2

2r2

+ (ln(r) + (θ + c1)2)F1(t) + F2(t) − 2ν ln(r)

t+c2

+

((θ+c1)2− 3

4 )r2

2(t+c2)2

.

JNCF Luminy 2018

• 2. Thomas decomposition of differential systems

JNCF Luminy 2018

Some references

• J. M. Thomas,

Differential Systems, AMS Colloquium Publications, vol. XXI, 1937.

• V. P. Gerdt,

On decomposition of algebraic PDE systems into simple subsystems, Acta Appl. Math., 101(1-3):39–51, 2008.

• T. B¨

achler, V. P. Gerdt, M. Lange-Hegermann, D. Robertz, Algorithmic Thomas Decomposition of Algebraic and Differential Systems, J. Symbolic Computation 47(10):1233–1266, 2012.

• D. Robertz,

Formal Algorithmic Elimination for PDEs, Lecture Notes in Mathematics, Vol. 2121, Springer, 2014.

JNCF Luminy 2018

Some references

• F. Boulier, D. Lazard, F. Ollivier, M. Petitot,

Representation for the radical of a finitely generated differential ideal, ISSAC 1995, pp. 158–166.

• D. Wang,

Decomposing polynomial systems into simple systems,

• J. Symbolic Computation 25(3):295–314, 1998.
• E. Hubert,

Notes on triangular sets and triangulation-decomposition algorithms. in: LNCS, Vol. 2630, 2003, pp. 1–39 and 40–87.

• F. Lemaire, M. Moreno Maza, Y. Xie,

The RegularChains library in Maple, SIGSAM Bulletin 39(3):96–97, 2005.

• D. Grigoriev,

Complexity of quantifier elimination in the theory of ordinary differential equations, in: LNCS, vol. 378, 1989, pp. 11–25.

JNCF Luminy 2018

Systems of PDEs

A differential system S is given by p1 = 0, p2 = 0, . . . , ps = 0, q1 = 0, q2 = 0, . . . , qt = 0, where p1, ..., ps and q1, ..., qt are polynomials in u1, ..., um of z1, ..., zn and their partial derivatives. Ω open and connected subset of Cn with coordinates z1, . . . , zn The solution set of S on Ω is SolΩ(S) := { f = (f1, . . . , fm) | fk : Ω → C analytic, k = 1, . . . , m, pi(f) = 0, qj(f) = 0, i = 1, . . . , s, j = 1, . . . , t } . Appropriate choice of Ω is possible only after formal treatment.

JNCF Luminy 2018

Systems of PDEs

A differential system S is given by p1 = 0, p2 = 0, . . . , ps = 0, q1 = 0, q2 = 0, . . . , qt = 0, Consequences of the system obtained in a finite number of steps from: p1 = 0, p2 = 0, . . . , ps = 0 are consequences, if p = 0 is consequence, then any partial derivative of p = 0 is, if p · q = 0 is consequence and q a factor of some qi, then p = 0 is consequence, if p = 0, r = 0 are consequences, then a p + b r = 0 is (a, b differential polynomials)

JNCF Luminy 2018

Polynomial ODEs / PDEs

du dt 2 − 4 t du dt − 4 u + 8 t2 = 0 find: u = u(t) analytic u(t) = a0 + a1 t + a2 t2

2! + a3 t3 3! + . . .

Substitute and compare coefficients:            a2

1 − 4 a0 = 0

2 a1 a2 − 8 a1 = 0 a1 a3 + a2

2 − 6 a2 + 8 = 0

. . . a0 := 0 ⇒ a1 = 0 ⇒ (a2 − 2)(a2 − 4) = 0 Many case distinctions? Thomas’ algorithm

• finitely many so-called simple systems

(Joseph Miller Thomas, ∼ 1930)

JNCF Luminy 2018

Algebraic geometry

L = { p1(x1, ..., xn) = 0, ..., pr = 0, q1 = 0, ..., qs = 0 } polynomial equations (and inequations) Sol(L) = { a ∈ Cn | pi(a) = 0, qj(a) = 0 ∀ i, j } Conversely, let S ⊆ Cn. I(S) = { p ∈ C[x1, ..., xn] | p(a) = 0 ∀ a ∈ S } Nullstellensatz (Hilbert, 1893) (for equations) radical ideals of C[x1, ..., xn] ← → zero sets in Cn are bijections which are inverse to each other.

JNCF Luminy 2018

Differential algebraic geometry

Differential algebra (Ritt, Kolchin, Seidenberg, . . . ) Q ⊆ K a differential field with commuting derivations ∂1, ..., ∂n Differential polynomial ring with derivations ∂1, ..., ∂n K{u} := K[∂i1

1 · · · ∂in n u | i ∈ (Z≥0)n] = K[u, uz1, ..., uzn, uz1,z1, ...]

K{u} not Noetherian (e.g., [u′u′′, u′′u′′′, . . .] ⊆ K{u} not fin. gen.)

• Thm. (Ritt-Raudenbush).

Every radical diff. ideal of K{u1, . . . , um} is finitely generated, is intersection of finitely many prime diff. ideals.

• Thm. (Differential Nullstellensatz).

Every radical diff. ideal I K{u1, . . . , um} has a zero in a diff. field ext.

• f K.

If f ∈ K{u1, . . . , um} vanishes for all zeros of I, then f ∈ I.

JNCF Luminy 2018

Thomas Decomposition

K{u} = K[u, ux, uy, . . . , ux,x, ux,y, uy,y, . . .]

• diff. polynomial ring

u < . . . < uy < ux < . . . < uy,y < ux,y < ux,x < . . . (ranking) algebraic reduction: p = u3

x,x,y + . . .

q = c u2

x,x,y + . . .

p → r = c · p − ux,x,y · q differential reduction: p = u3

x,x,y,y + . . .

q = c u2

x,x,y + . . .

∂y q =

∂q ∂ux,x,y ux,x,y,y + . . .

p → r =

∂q ∂ux,x,y · p − u2 x,x,y,y · ∂y q

reduction requires: initial c = 0 and separant

∂q ∂ux,x,y = 0

JNCF Luminy 2018

Thomas Decomposition

R = K{u1, . . . , um} Def. Thomas decomposition

• f diff. system S

(or Sol(S)): Sol(S) = Sol(S1) ⊎ . . . ⊎ Sol(Sr), Si simple diff. system Thm. S = {p1 = 0, ..., ps = 0, q1 = 0, ..., qt = 0} simple diff. system E

• diff. ideal generated by p1, . . . , ps

q product of initials and separants of all pi Then E : q∞ := { p ∈ R | qr · p ∈ E for some r ∈ Z≥0 } = IR(Sol(S)) consists of all diff. polynomials in R vanishing on Sol(S).

JNCF Luminy 2018

Thomas Decomposition

p = x3 + (3y + 1)x2 + (3y2 + 2y)x + y3 = 0

x y

JNCF Luminy 2018

Thomas Decomposition

p = x3 + (3y + 1)x2 + (3y2 + 2y)x + y3 = 0

x y

JNCF Luminy 2018

Thomas Decomposition

p = x3 + (3y + 1)x2 + (3y2 + 2y)x + y3 = 0

x y

discx(p) = y2(4 − 27y2)

JNCF Luminy 2018

Thomas Decomposition

p = x3 + (3y + 1)x2 + (3y2 + 2y)x + y3 = 0

x y 2 non-real points

discx(p) = y2(4 − 27y2)

JNCF Luminy 2018

Simple Systems

K field of char. 0, p1, . . . , ps, q1, . . . , qt ∈ K[x1, . . . , xn] V =

• a ∈ K

n

pi(a) = 0, qj(a) = 0 ∀ i, j

• πi : K

n−(i−1) −

→ K

n−i : (ai, ai+1, . . . , an) −

→ (ai+1, . . . , an) V1 := V , Vi+1 := πi(Vi) V is simple, if for each i

• ne of the following three cases holds:

           ∃ e ∀(ai+1, . . . , an) ∈ πi(Vi) ∃! a(1)

i , . . . , a(e) i

(a(j)

i , ai+1, . . . , an) ∈ Vi,

∃ f ∀(ai+1, . . . , an) ∈ πi(Vi) ∃! a(1)

i , . . . , a(f) i

(a(j)

i , ai+1, . . . , an) ∈ Vi,

∀(ai+1, . . . , an) ∈ πi(Vi) (ai, ai+1, . . . , an) ∈ Vi ∀ai ∈ K Thomas decomposition: Write V = W1 ⊎ . . . ⊎ Wr where Wj simple

JNCF Luminy 2018

Simple Systems

p1, . . . , ps, q1, . . . , qt ∈ K[x1, . . . , xn], x1 > x2 > . . . > xn V =

• a ∈ K

n

pi(a) = 0, qj(a) = 0 ∀ i, j

• Identify

K[x1, . . . , xn] = K[xn][xn−1] . . . [x1]. S = {p1 = 0, . . . , ps = 0, q1 = 0, . . . , qt = 0} is a simple system, if

• 1. Each variable is leader of at most one

pi

• r

qj.

• 2. The initial of

pi, qj has no zero in πk(Vk), if xk is the leader of pi resp. qj.

• 3. pi(xk, ak+1, . . . , an),

qj(xk, ak+1, . . . , an) are square-free for all (ak+1, . . . , an) ∈ πk(Vk), if xk is the leader of pi resp. qj.

JNCF Luminy 2018

Thomas Decomposition

p = ax2 + bx + c = 0, p ∈ Q[x, c, b, a], x > c > b > a ax2 + bx + c = 0

JNCF Luminy 2018

Thomas Decomposition

p = ax2 + bx + c = 0, p ∈ Q[x, c, b, a], x > c > b > a ax2 + bx + c = 0 a = 0 bx + c = 0 a = 0

JNCF Luminy 2018

Thomas Decomposition

p = ax2 + bx + c = 0, p ∈ Q[x, c, b, a], x > c > b > a ax2 + bx + c = 0 4ac − b2 = 0 a = 0 2ax + b = 0 4ac − b2 = 0 a = 0 bx + c = 0 a = 0

JNCF Luminy 2018

Thomas Decomposition

p = ax2 + bx + c = 0, p ∈ Q[x, c, b, a], x > c > b > a ax2 + bx + c = 0 4ac − b2 = 0 a = 0 2ax + b = 0 4ac − b2 = 0 a = 0 bx + c = 0 b = 0 a = 0 c = 0 b = 0 a = 0

JNCF Luminy 2018

Thomas Decomposition

p = ax2 + bx + c = 0, p ∈ Q[x, c, b, a], x > c > b > a ax2 + bx + c = 0 4ac − b2 = 0 a = 0 2ax + b = 0 4ac − b2 = 0 a = 0 bx + c = 0 b = 0 a = 0 c = 0 b = 0 a = 0 x1 = x2 x1 = x2 x1 all x ∈ Q solve p(x) = 0 for fixed choice of a, b, c

JNCF Luminy 2018

Thomas Decomposition

• x2 + y2 − 1

= (1 − y) t − x =

(x, y) t

p1 := x2 + y2 − 1, p2 := x + t y − t ∈ Q[x, y, t], x > y > t p1 − (x − ty + t) p2 = (y − 1) ((t2 + 1)y − t2 + 1) Thomas decomposition: (t2 + 1) x − 2t = 0 (t2 + 1) y − t2 + 1 = 0 t2 + 1 = 0 x = 0 y − 1 = 0

JNCF Luminy 2018

Thomas Decomposition

S = { p1 = 0, . . . , ps = 0, q1 = 0, . . . , qt = 0 } Def. Thomas decomposition

• f diff. system S

(or Sol(S)): Sol(S) = Sol(S1) ⊎ . . . ⊎ Sol(Sr), Si simple diff. system Def. S is simple if (a) p1, . . . , ps, q1, . . . , qt have pairwise distinct leaders, (b) leading coefficients and discriminants of pi and qj do not vanish, (c) p1, . . . , ps form a passive PDE system, (d) q1, . . . , qt are reduced modulo p1, . . . , ps. set of admissible derivations µi ⊆ {∂1, . . . , ∂n} for pi, i = 1, . . . , s

JNCF Luminy 2018

Thomas Decomposition

p = ˙ u2 − 4t ˙ u − 4u + 8t2 = 0 p ∈ Q(t){u} Separant of p:

∂p ∂ ˙ u = 2 ˙

u − 4t res(p, ∂p

∂ ˙ u, ˙

u) = −16u + 16t2 Thomas decomposition: p = 0 u − t2 = 0 u − t2 = 0 general solution: u(t) = 2((t + c)2 + c2), c ∈ R essential singular solution: u(t) = t2

JNCF Luminy 2018

Thomas Decomposition

       ∂2u ∂x2 − ∂2u ∂y2 = 0, ∂u ∂x − u2 = Define p1 := ux,x − uy,y, p2 := ux − u2 R = Q{u} with commuting derivations ∂x, ∂y. degree-reverse lexicographical ranking > on R with ∂x u > ∂y u p3 := p1 − ∂x p2 − 2 u p2 = −uy,y + 2 u3 . Janet division associates the sets of admissible derivations:    ux − u2 = 0, {∂x, ∂y} uy,y − 2 u3 = 0, { ∗ , ∂y}

JNCF Luminy 2018

Thomas Decomposition

passivity check: ∂x p3 + ∂2

y p2 − 6 u2 p2 − 2 u p3

= −2 (uy2 − u4) = −2 (uy + u2) (uy − u2) . factorization

• splitting possible

If the above factorization is ignored, then the discriminant of p4 := u2

y − u4 needs to be considered, which implies vanishing or

non-vanishing of the separant 2 uy. This case distinction leads to the Thomas decomposition ux − u2 = 0, {∂x, ∂y} uy2 − u4 = 0, { ∗ , ∂y} u = u =

JNCF Luminy 2018

Thomas Decomposition

passivity check: ∂x p3 + ∂2

y p2 − 6 u2 p2 − 2 u p3

= −2 (uy2 − u4) = −2 (uy + u2) (uy − u2) . factorization

• splitting possible

For both systems a differential reduction of p3 modulo the chosen factor is applied because the monomial ∂y defining the new leader divides the monomial ∂y,y defining ld(p3). We obtain the Thomas decomposition ux − u2 = 0, {∂x, ∂y} uy + u2 = 0, { ∗ , ∂y} ux − u2 = 0, {∂x, ∂y} uy − u2 = 0, { ∗ , ∂y} u = 0.

JNCF Luminy 2018

Thomas Decomposition

∂u ∂t − 6 u ∂u ∂x + ∂3u ∂x3 = 0, u ∂2u ∂t∂x − ∂u ∂t ∂u ∂x = 0, R := K{u} with commuting derivations ∂t, ∂x degree-reverse lexicographical ranking on R with ut > ux Define p := ut − 6 u ux + ux,x,x, q := u ut,x − ut ux ld(p) = ux,x,x, ld(q) = ut,x, init(p) = 1, and init(q) = u. Hence, { p = 0, q = 0 } is a triangular set. We replace this system with two systems { p = 0, q = 0, u = 0 }, { p = 0, q = 0, u = 0 } according to vanishing or non-vanishing initial of q.

JNCF Luminy 2018

Thomas Decomposition

The first system is equivalent to the simple differential system S1 := { u = 0 }. The second system is simple as an algebraic system, but not passive. Janet division associates the sets of admissible derivations:

• p := ut − 6 u ux + ux,x,x ,

{ ∗ , ∂x} q := u ut,x − ut ux , {∂t , ∂x} Janet reduction of ∂t p modulo { (p, {∂x}), (q, {∂t, ∂x}) } yields r := u (u pt − qx,x) − u ut p + ux qx = u2 ut,t − u (6 u2 − ux,x) ut,x − ut ux ux,x − u u2

t.

JNCF Luminy 2018

Thomas Decomposition

{ p = 0, q = 0, r = 0, u = 0 } is simple as an algebraic system. Janet reduction of ∂t q modulo { (p, {∂x}), (q, {∂x}), (r, {∂t, ∂x}) }: s := u ((u qt − rx) − (6 u2 − ux,x) qx + q p) + 3 ux r + 3 (2 u2 ux − u ut − ux ux,x) q = 6 u3 ut ux,x. We have init(s) = 6 u3 ut. Now, init(s) = 0 implies ux,x = 0, which results in the simple system S2 := { ut − 6 u ux = 0, ux,x = 0, u = 0 }. On the other hand, init(s) = 0 implies ut = 0, hence the simple system S3 := { ut = 0, ux,x,x − 6 u ux = 0, ux,x = 0, u = 0 }. Thomas decomposition: S1, S2, S3

JNCF Luminy 2018

Thomas Decomposition

ut − 6uux + ux,x,x = 0 (Korteweg-de Vries) Ansatz: u(t, x) = f(t)g(x) PDE: uut,x − utux = 0 Thomas decomposition of {ut − 6uux + ux,x,x = 0, uut,x − utux = 0}: u = 0 ut − 6uux = 0 ux,x = 0 u = 0 ut = 0 ux,x,x − 6uux = 0 ux,x = 0 u = 0 solutions: u(t, x) =

x+c1 −6t+c2

x = ± u(x)

u(0)

dz √ 2z3 − az − b

JNCF Luminy 2018

Thomas Decomposition

1937: J. M. Thomas: “Differential Systems”. 1998: D. Wang: implementation for algebraic systems 2007: V. Gerdt: “On decomposition of algebraic PDE systems into simple subsystems” 2009: W. Plesken: “Counting solutions of polynomial systems via iterated fibrations” since 2009: implementations in Maple for algebraic systems (T. B¨ achler) systems of PDEs (M. Lange-Hegermann)

• T. B¨

achler, V. P. Gerdt, M. Lange-Hegermann, D. R., Algorithmic Thomas decomposition of algebraic and differential systems,

• J. Symbolic Computation 47(10):1233–1266, 2012.

JNCF Luminy 2018

Example

det   ux,x ux,y uy,y ux,y uy,y uy,z ux,z uy,z uz,z   = 0, Q{u} with degrevlex ranking Thomas decomposition:

det(. . .) = uz,zuy,y − u2

y,z

= uz,z = −u2

y,zux,x + 2uy,zux,zux,y − uy,yu2 x,z

= uy,z = uz,z = uz,zux,y − uy,zux,z = uz,zuy,y − u2

y,z

= uz,z = ux,z = uy,z = uz,z = uy,y = uy,y = uy,z = uz,z =

JNCF Luminy 2018

Example

det   ux,x ux,y uy,y ux,y uy,y uy,z ux,z uy,z uz,z   = 0, Q{u} with degrevlex ranking

det(. . .) = 0 (T1) (T2) (T3) (T4) (T5) uz,z uy,y − u2

y,z = 0

uz,z uy,y − u2

y,z = 0

uz,z = 0 uz,z = 0 uz,z = 0 uz,z = 0 uy,y = 0 uy,y = 0

JNCF Luminy 2018

Example

det   ux,x ux,y uy,y ux,y uy,y uy,z ux,z uy,z uz,z   = 0, Q{u} with degrevlex ranking DifferentialAlgebra (Maple 17):

det(. . .) = uz,zuy,y − u2

y,z

= uz,zux,y − uy,zux,z = uz,zuy,y − u2

y,z

= uz,z = ux,z = uy,z = uz,z = uy,y = uy,z = uz,z =

JNCF Luminy 2018

Thomas Decomposition

R = K{u1, . . . , um} Thm. S = {p1 = 0, ..., ps = 0, q1 = 0, ..., qt = 0} simple diff. system E

• diff. ideal generated by p1, . . . , ps,

q product of initials and separants of all pi. Then E : q∞ := { p ∈ R | qr · p ∈ E for some r ∈ Z≥0 } = IR(Sol(S)) consists of all diff. polynomials in R vanishing on Sol(S). Thm. S not necessarily simple S1, . . . , Sr Thomas decomposition of S w.r.t. any ranking on R √E : q∞ =

• E(1) : (q(1))∞

∩ . . . ∩

• E(r) : (q(r))∞

JNCF Luminy 2018

Elimination

Lemma

J ⊆ R := K[X1, . . . , Xn, Y1, . . . , Ym] Janet basis w.r.t. any term order. For any 0 = p ∈ R let lm(p) be its leading monomial. If { p ∈ J | p ∈ K[Y1, . . . , Ym] } = { p ∈ J | lm(p) ∈ K[Y1, . . . , Ym] }, then J ∩ K[Y1, . . . , Ym] generates J ∩ K[Y1, . . . , Ym]. Proof. Let 0 = p ∈ J ∩ K[Y1, . . . , Ym]. Since J is a Janet basis, ∃ q ∈ J, lm(q) ∈ K[Y1, . . . , Ym], lm(q) | lm(p). By assumption, q ∈ K[Y1, . . . , Ym]. Reduction p → 0 in K[Y1, . . . , Ym].

• JNCF Luminy 2018

Differential Elimination

Lemma

Let S be simple, w.r.t. any ranking >, E

• diff. ideal generated by

S= = { p1, . . . , ps }, q

• prod. init. sep. of all pi,

V ⊂ { u1, . . . , um } If P := { p ∈ S= | p ∈ K{V } } = { p ∈ S= | ld(p) ∈ Mon(∆) V }, then (E : q∞) ∩ K{V } = E′ : (q′)∞, E′

• diff. ideal of K{V } gen. by P,

q′

• prod. of init. and sep. of p ∈ P.

Proof. Let 0 = p ∈ (E : q∞) ∩ K{V }. Since S is simple, b p = R-linear comb. of p1, . . . , ps and their derivatives By assumption, for every Janet divisor is in K{V }. Pseudo-reduction p → 0 in K{V }.

• JNCF Luminy 2018

Nonlinear control systems

Stirred tank:    ˙ V (t) = F1(t) + F2(t) − k

• V (t)

˙ c(t) V (t) = c1 F1(t) + c2 F2(t) − c(t) k

• V (t)
• H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, John Wiley & Sons, 1972.

JNCF Luminy 2018

Nonlinear control systems

2-D crane:            m ¨ x = −T sin θ m ¨ z = −T cos θ + m g x = R sin θ + d z = R cos θ

• M. Fliess, J. L´

evine, P. Martin, P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples, Internat. J. Control 61(6), 1327–1361, 1995.

JNCF Luminy 2018

Nonlinear control systems

Unicycle:      ˙ x1 = cos(x3) u1 ˙ x2 = sin(x3) u1 ˙ x3 = u2

• H. Nijmeijer, A. van der Schaft, Nonlinear dynamical control systems, Springer, 1990.
• G. Conte, C. H. Moog, A. M. Perdon, Nonlinear control systems, Vol. 242 of LNCIS,

Springer, 1999.

JNCF Luminy 2018

Algebraic approach to systems theory

A few references:

• S. Diop,

Elimination in control theory,

• Math. Control Signals Systems 4(1):17–32, 1991.
• S. Diop,

Differential-algebraic decision methods and some applications to system theory,

• Theoret. Comput. Sci. 98(1):137–161, 1992.
• M. Fliess, S. T. Glad,

An Algebraic Approach to Linear and Nonlinear Control, in: H. L. Trentelman and J. C. Willems (eds.), Essays on Control: Perspectives in the Theory and its Applications,

• pp. 223–267, Birkh¨

auser, 1993.

• M. Fliess, J. L´

evine, P. Martin, P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples,

• Internat. J. Control 61(6):1327–1361, 1995.

J.-F. Pommaret, Partial differential control theory, Kluwer, 2001.

JNCF Luminy 2018

Example

Unicycle:      ˙ x1 = cos(x3) u1 ˙ x2 = sin(x3) u1 ˙ x3 = u2

• H. Nijmeijer, A. van der Schaft, Nonlinear dynamical control systems, Springer, 1990.
• G. Conte, C. H. Moog, A. M. Perdon, Nonlinear control systems, Vol. 242 of LNCIS,

Springer, 1999.

JNCF Luminy 2018

Example

R = Q{x1, x2, cx3, sx3, u1, u2, y1, y2} block ranking > with {x1, x2, cx3, sx3} ≫ {u1, u2} ≫ {y1, y2}

> with(DifferentialThomas): > ivar := [t]:

dvar := [x1,x2,cx3,sx3,u1,u2,y1,y2]:

> ComputeRanking(ivar,

[[x1,x2,cx3,sx3],[u1,u2],[y1,y2]]):

> L := [x1[t]-cx3*u1, x2[t]-sx3*u1, sx3[t]-cx3*u2,

y1-x1, y2-x2, cx3^2+sx3^2-1]:

> LL := Diff2JetList(Ind2Diff(L, ivar, dvar));

LL := [x1 1 − cx3 0u1 0, x2 1 − sx3 0u1 0, sx3 1 − cx3 0u2 0, y1 0 − x1 0, y2 0 − x2 0, cx3 02 + sx3 02 − 1]

> TD := DifferentialThomasDecomposition(LL, [cx3]);

TD := [DifferentialSystem, DifferentialSystem, DifferentialSystem, DifferentialSystem, DifferentialSystem]

JNCF Luminy 2018

Example

> Print(TD[1]);

[x1 − y1 = 0, x2 − y2 = 0, u1 cx3 − y1 t = 0, u1 sx3 − y2 t = 0, u1 2 − y1 t

2 − y2 t 2 = 0,

y1 t

2 u2 + y2 t 2u2 − y1 ty2 t,t + y2 ty1 t,t = 0,

y2 t = 0, y1 t = 0, y1 t

2 + y2 t 2 = 0,

y2 ty1 t,t − y1 ty2 t,t = 0]

> collect(%[6], u2, factor);

• y1 t

2 + y2 t 2

u2 − y1 ty2 t,t + y2 ty1 t,t = 0

> Print(TD[2]);

[x1 − y1 = 0, x2 − y2 = 0, u1 cx3 − y1 t = 0, u1 sx3 − y2 t = 0, u1 2 − y1 t

2 − y2 t 2 = 0,

u2 = 0, y2 ty1 t,t − y1 ty2 t,t = 0, y2 t = 0, y1 t = 0, y1 t

2 + y2 t 2 = 0]

• (y1)t

(y2)t (y1)t,t (y2)t,t

• = 0

⇒ ˙ x1 and ˙ x2 are proportional

JNCF Luminy 2018

Example

> Print(TD[3]);

[x1 − y1 = 0, x2 − y2 = 0, cx3 + 1 = 0, sx3 = 0, u1 + y1 t = 0, u2 = 0, y2 t = 0, y1 t = 0]

> Print(TD[4]);

[x1 − y1 = 0, x2 − y2 = 0, cx3 − 1 = 0, sx3 = 0, u1 − y1 t = 0, u2 = 0, y2 t = 0, y1 t = 0] movement restricted to any of the two directions defined by the x1-coordinate, no rotation allowed

> Print(TD[5]);

[x1 − y1 = 0, x2 − y2 = 0, cx3 2 + sx3 2 − 1 = 0, sx3 t − u2 cx3 = 0, u1 = 0, y1 t = 0, y2 t = 0, sx3 + 1 = 0, sx3 − 1 = 0]

• nly rotation allowed, u1 is zero function

JNCF Luminy 2018

Control Theory

R = K{u1, . . . , um}, U := {u1, . . . , um}, S simple diff. system Def. x ∈ U is observable w.r.t. Y ⊆ U − {x} ⇐ ⇒            ∃ p ∈ (E : q∞) − {0} s.t. p ∈ K{Y }[x] (without derivatives of x) initial of p ∈ (E : q∞), ∂p ∂x ∈ (E : q∞) Def. Y ⊆ U is a flat output ⇐ ⇒    (E : q∞) ∩ K{Y } = {0} every x ∈ U − Y is observable w.r.t. Y

JNCF Luminy 2018

Example

Stirred tank:    ˙ V (t) = F1(t) + F2(t) − k

• V (t)

˙ c(t) V (t) = c1 F1(t) + c2 F2(t) − c(t) k

• V (t)
• H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, John Wiley & Sons, 1972.

JNCF Luminy 2018

Example

R = Q{F1, F2, sV, c, c1, c2}, ranking > s.t. {F2, F2} ≫ {sV, c} ≫ {c1, c2}

> with(DifferentialThomas): > ivar := [t]:

dvar := [F1,F2,sV,c,c1,c2]:

> ComputeRanking(ivar, [[F1,F2],[sV,c],[c1,c2]]): > L := [2*sV[t]*sV-F1-F2+k*sV,

c[t]*sV^2-c2*F2+c*k*sV-c1*F1+2*c*sV[t]*sV, c1[t], c2[t]]:

> LL := Diff2JetList(Ind2Diff(L, ivar, dvar));

LL := [2 sV 1sV 0 − F1 0 − F2 0 + ksV 0, c1sV 0

2 − c2 0F2 0 + c0ksV 0 − c1 0F1 0 + 2 c0sV 1sV 0,

c1 1, c2 1]

> TD := DifferentialThomasDecomposition(LL,

[sV[0],c1[0],c2[0]]); TD := [DifferentialSystem, DifferentialSystem, DifferentialSystem]

JNCF Luminy 2018

Example

> Print(TD[1]);

[c2 F1 − c1 F1 + 2 csV sV t − 2 c2 sV sV t + ctsV 2 + cksV − c2 ksV = 0, c1 F2 − c2 F2 + 2 csV sV t − 2 c1 sV sV t + ctsV 2 + cksV − c1 ksV = 0, c1 t = 0, c2 t = 0, c2 = 0, c1 = 0, c1 − c2 = 0, sV = 0]

> collect(%[1], F1);

(c2 − c1) F1 + 2 csV sV t − 2 c2 sV sV t + ctsV 2 + cksV − c2 ksV = 0

> collect(%%[2], F2);

(c1 − c2) F2 + 2 csV sV t − 2 c1 sV sV t + ctsV 2 + cksV − c1 ksV = 0 ⇒ F1, F2 observable with respect to {c, sV } (E : q∞) ∩ Q{sV, c} = {0} ⇒ {c, sV } is flat output

JNCF Luminy 2018

Example

> Print(TD[2]);

[cF1 − c2 F1 + cF2 − c2 F2 + ctsV 2 = 0, 2 csV t − 2 c2 sV t + ctsV + ck − c2 k = 0, c1 − c2 = 0, c2 t = 0, c2 = 0, c − c2 = 0, sV = 0]

> Print(TD[3]);

[F1 + F2 − 2 sV sV t − ksV = 0, c − c2 = 0, c1 − c2 = 0, c2 t = 0, c2 = 0, sV = 0] conditions c1 = c2 and (c1)t = (c2)t = 0 preclude control of the concentration in the tank

JNCF Luminy 2018

Example

2-D crane:            m ¨ x = −T sin θ m ¨ z = −T cos θ + m g x = R sin θ + d z = R cos θ

• M. Fliess, J. L´

evine, P. Martin, P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples, Internat. J. Control 61(6), 1327–1361, 1995.

JNCF Luminy 2018

Example

Q(m, g){T, s, c, d, R, x, z} block ranking > satisfying {T, s, c, d, R} ≫ {x, z}

> with(DifferentialThomas): > ivar := [t]:

dvar := [T,s,c,d,R,x,z]:

> ComputeRanking(ivar, [[T,s,c,d,R],[x,z]]): > TD := DifferentialThomasDecomposition(

[m*x[2]+T[0]*s[0], m*z[2]+T[0]*c[0]-m*g, x[0]-R[0]*s[0]-d[0], z[0]-R[0]*c[0], c[0]^2+s[0]^2-1], []); TD := [DifferentialSystem, DifferentialSystem, DifferentialSystem, DifferentialSystem, DifferentialSystem, DifferentialSystem, DifferentialSystem]

JNCF Luminy 2018

Example

> Print(TD[2]);

[zT + mzt,tR − mgR = 0, zt,tRs − gRs − zxt,t = 0, Rc − z = 0, zt,td − gd + zxt,t − xzt,t + gx = 0, zt,t2R2 − 2 gzt,tR2 + g2R2 − z2xt,t2 − z2zt,t2 + 2 gz2zt,t − g2z2 = 0, z = 0, zt,t − g = 0, xt,t = 0, xt,t2 + zt,t2 − 2 gzt,t + g2 = 0]

> collect(%[5], R, factor);

(zt,t − g)2 R2 − z2 xt,t

2 + zt,t 2 − 2 gzt,t + g2

= 0 ⇒ T, s, c, d, R observable with respect to {x, z} (E : q∞) ∩ Q{x, z} = {0} ⇒ {x, z} is flat output

JNCF Luminy 2018

Example

> Print(TD[1]);

[T = 0, Rs + d − x = 0, Rc − z = 0, d2 − 2 xd + x2 − R2 + z2 = 0, xt,t = 0, zt,t − g = 0, z = 0, R = 0, R + z = 0, R − z = 0]

> Print(TD[3]);

[T − mzt,t + mg = 0, s = 0, c + 1 = 0, d − x = 0, R + z = 0, xt,t = 0, z = 0]

> Print(TD[4]);

[T + mzt,t − mg = 0, s = 0, c − 1 = 0, d − x = 0, R − z = 0, xt,t = 0, z = 0]

> Print(TD[5]);

[cT − mg = 0, gs + xt,tc = 0, g2c2 + xt,t2c2 − g2 = 0, d − x = 0, R = 0, z = 0, xt,t = 0, xt,t2 + g2 = 0]

> Print(TD[6]);

[T + mg = 0, s = 0, c + 1 = 0, d − x = 0, R = 0, xt,t = 0, z = 0]

> Print(TD[7]);

[T − mg = 0, s = 0, c − 1 = 0, d − x = 0, R = 0, xt,t = 0, z = 0]

JNCF Luminy 2018

Example

                 −a(x2) ∂ξ1(x) ∂x1 + ∂ξ3(x) ∂x1 − ∂ ∂x2 a(x2)

• ξ2(x) + 1

2 a(x2) (∇ · ξ(x)) = −a(x2) ∂ξ1(x) ∂x2 + ∂ξ3(x) ∂x2 = −a(x2) ∂ξ1(x) ∂x3 + ∂ξ3(x) ∂x3 − 1 2 (∇ · ξ(x)) =

infinitesimal transformations of a Pfaffian system ∂ ∂x1 a(x1, x2, x3) = 0, ∂ ∂x3 a(x1, x2, x3) = 0.

J.-F. Pommaret, A. Quadrat, Formal obstructions to the controllability of partial differential control systems, Proc. IMACS, Berlin, vol. 5, pp. 209–214, 1997.

JNCF Luminy 2018

Example

R = Q(x1, x2, x3){ξ1, ξ2, ξ3, a} with diff. op. ∂1, ∂2, ∂3 w.r.t. x1, x2, x3 block ranking > on R with blocks {ξ1, ξ2, ξ3}, {a}

> with(DifferentialThomas): > ivar := [x1,x2,x3]:

dvar := [xi1,xi2,xi3,a]:

> ComputeRanking(ivar, [[xi1,xi2,xi3],[a]]): > L := [-a*xi1[x1]+xi3[x1]-a[x2]*xi2

+(1/2)*a*(xi1[x1]+xi2[x2]+xi3[x3]), -a*xi1[x2]+xi3[x2],

• a*xi1[x3]+xi3[x3] -(1/2)*(xi1[x1]+xi2[x2]+xi3[x3]),

a[x1], a[x3]]:

> LL := Diff2JetList(Ind2Diff(L, ivar, dvar));

LL := [−a0,0,0ξ1 1,0,0 + ξ3 1,0,0 + 1/2 a0,0,0

• ξ1 1,0,0 + ξ2 0,1,0 + ξ3 0,0,1
• −a0,1,0ξ2 0,0,0,

−a0,0,0ξ1 0,1,0 + ξ3 0,1,0, −a0,0,0ξ1 0,0,1 + 1/2 ξ3 0,0,1 − 1/2 ξ1 1,0,0 − 1/2 ξ2 0,1,0, a1,0,0, a0,0,1]

> TD := DifferentialThomasDecomposition(LL, []);

TD := [DifferentialSystem, DifferentialSystem, DifferentialSystem]

JNCF Luminy 2018

Example

> Print(TD[1]);

[aξ1 x2 − ξ3 x2 = 0, a2ξ1 x3 + ξ3 x1 = 0, ξ2 = 0, aξ1 x1 − 2 ξ3 x1 − aξ3 x3 = 0, ax1 = 0, ax3 = 0, a = 0] parameter a = a(x2) non-zero, but otherwise arbitrary

> Print(TD[2]);

[aξ1 x2 − ξ3 x2 = 0, a2ξ1 x3 + aξ2 x2 − ax2ξ2 + ξ3 x1 = 0, aξ1 x1 − aξ2 x2 + 2 ax2ξ2 − 2 ξ3 x1 − aξ3 x3 = 0, ax1 = 0, ax2,x2 = 0, ax2,x3 = 0, ax3 = 0, a = 0, ξ2 = 0] parameter subject to ax2,x2 = 0

> Print(TD[3]);

[ξ1 x1,x1 + ξ2 x1,x2 = 0, ξ1 x1,x2 + ξ2 x2,x2 = 0, ξ3 x1 = 0, ξ3 x2 = 0, ξ1 x1 + ξ2 x2 − ξ3 x3 = 0, a = 0] parameter a = 0

JNCF Luminy 2018

References

• J. M. Thomas,

Differential Systems, AMS Colloquium Publications, vol. XXI, 1937.

• J. M. Thomas,

Systems and Roots, William Byrd, Richmond, 1962.

• A. Seidenberg,

An elimination theory for differential algebra,

• Univ. California Publ. Math. (N.S.), 3:31–65, 1956.
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Complexity of quantifier elimination in the theory of ordinary differentially closed fields,

• J. Soviet Math. 59(3):814–822, 1992.
• V. P. Gerdt,

On decomposition of algebraic PDE systems into simple subsystems, Acta Appl. Math., 101(1-3):39–51, 2008.

JNCF Luminy 2018

References

Wu W.-T., Mathematics mechanization, Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equtaions-solving, Kluwer, 2000.

• D. Wang,

Decomposing polynomial systems into simple systems,

• J. Symbolic Comput. 25(3):295–314, 1998.
• Z. Li, D. Wang,

Coherent, regular and simple systems in zero decompositions of partial differential systems,

• Syst. Sci. Math. Sci. 12(Suppl.):43–60, 1999.
• D. Wang,

Elimination practice. Software tools and applications, Imperial College Press, London, 2004.

JNCF Luminy 2018

References

• F. Boulier, D. Lazard, F. Ollivier, M. Petitot,

Representation for the radical of a finitely generated differential ideal, ISSAC 1995, pp. 158–166.

• E. Hubert,

Notes on triangular sets and triangulation-decomposition algorithms.

• I. Polynomial systems. II. Differential systems,

in: F. Winkler, U. Langer (eds.), LNCS, Vol. 2630, Springer, 2003.

• F. Lemaire, M. Moreno Maza, Y. Xie,

The RegularChains library in Maple, SIGSAM Bulletin 39(3):96–97, 2005.

• P. Aubry, D. Lazard, M. Moreno Maza,

On the theories of triangular sets,

• J. Symbolic Comp. 28(1–2):105–124, 1999.

JNCF Luminy 2018

References

• W. Plesken,

Counting solutions of polynomial systems via iterated fibrations,

• Arch. Math. (Basel), 92(1):44–56, 2009.
• W. Plesken,

Gauss-Bruhat decomposition as an example of Thomas decomposition,

• Arch. Math. (Basel), 92(2):111–118, 2009.
• T. B¨

achler, V. P. Gerdt, M. Lange-Hegermann, D. Robertz, Algorithmic Thomas Decomposition of Algebraic and Differential Systems,

• J. Symbolic Comp. 47(10):1233–1266, 2012.
• T. B¨

achler, V. P. Gerdt, M. Lange-Hegermann, D. Robertz, Thomas Decomposition of Algebraic and Differential Systems, In V. P. Gerdt, W. Koepf, E. W. Mayr, and E. H. Vorozhtsov (eds.), Computer Algebra in Scientific Computing, 2010, Tsakhkadzor, Armenia, LNCS, Vol. 6244, pp. 31–54. Springer, 2010.

JNCF Luminy 2018

References

• D. Robertz,

Formal Algorithmic Elimination for PDEs, Lecture Notes in Mathematics, Vol. 2121, Springer, 2014.

• M. Lange-Hegermann, D. Robertz,

Thomas Decomposition and Nonlinear Control Systems, accepted for publication.

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Thomas decompositions of parametric nonlinear control systems,

• Proc. 5th Symposium on System Structure and Control, Grenoble

(France), pp. 291–296, 2013.

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Counting Solutions of Differential Equations, PhD thesis, RWTH Aachen University, Germany, 2014. Available at http://publications.rwth-aachen.de/record/229056.

JNCF Luminy 2018

References

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Differential algebraic modelling of nonlinear systems, in: M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran (eds.), Realization and Modelling in System Theory, 97–105, Birkh¨ auser, 1989.

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Differential-algebraic decision methods and some applications to system theory, Theoret. Comput. Sci. 98(1):137–161, 1992.

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Radon Ser. Comput. Appl. Math., Vol. 3, pp. 127–149, de Gruyter, 2007.

JNCF Luminy 2018

References

• M. Fliess, S. T. Glad,

An Algebraic Approach to Linear and Nonlinear Control, in: H. L. Trentelman and J. C. Willems (eds.), Essays on Control: Perspectives in the Theory and its Applications, pp. 223–267, Birkh¨ auser, 1993.

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evine, P. Martin, P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples,

• Internat. J. Control 61(6):1327–1361, 1995.

J.-F. Pommaret, Partial differential control theory, Kluwer, 2001.

JNCF Luminy 2018

References

• H. Kwakernaak, R. Sivan,

Linear Optimal Control Systems, John Wiley & Sons, 1972. J.-F. Pommaret, A. Quadrat Formal obstructions to the controllability of partial differential control systems, in: Proc. IMACS, Berlin, Vol. 5, pp. 209–214, 1997.

JNCF Luminy 2018

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eray, D´ emonstration g´ en´ erale de l’existence des int´ egrales des ´ equations aux d´ eriv´ ees partielles,

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Differential Systems, AMS Colloquium Publications, vol. XXI, 1937.

JNCF Luminy 2018

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Differential Algebra, Dover, 1966.

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Differential algebra and algebraic groups, Academic Press, 1973.

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An elimination theory for differential algebra,

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JNCF Luminy 2018

References

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Linear Differential Elimination for Analytic Functions, Mathematics in Computer Science, 4(2–3):231–242, 2010.

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Elimination for coefficients of special characteristic polynomials, Experimental Mathematics, 17(4):499–510, 2008.

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Janet’s approach to presentations and resolutions for polynomials and linear pdes, Archiv der Mathematik, 84(1):22–37, 2005.

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Computer Algebra in Scientific Computing 2003.

JNCF Luminy 2018

References

• D. Robertz,

Formal Algorithmic Elimination for PDEs, Lecture Notes in Mathematics, Vol. 2121, Springer, 2014.

• T. B¨

achler, V. P. Gerdt, M. Lange-Hegermann, D. Robertz, Algorithmic Thomas Decomposition of Algebraic and Differential Systems, J. Symbolic Computation 47(10):1233–1266, 2012.

• T. B¨

achler, V. P. Gerdt, M. Lange-Hegermann, D. Robertz, Thomas Decomposition of Algebraic and Differential Systems, In V. P. Gerdt, W. Koepf, E. W. Mayr, and E. H. Vorozhtsov (eds.), Computer Algebra in Scientific Computing 2010, Tsakhkadzor, Armenia, LNCS, Vol. 6244, 2010, pp. 31–54.

• V. P. Gerdt, D. Robertz,

Lagrangian Constraints and Differential Thomas Decomposition, Advances in Applied Mathematics 72:113–138, 2016.

JNCF Luminy 2018

References

• V. P. Gerdt,

On decomposition of algebraic PDE systems into simple subsystems, Acta Appl. Math., 101(1-3):39–51, 2008.

• W. Plesken,

Counting solutions of polynomial systems via iterated fibrations,

• Arch. Math. 92(1):44–56, 2009.
• W. Plesken,

Gauss-Bruhat decomposition as an example of Thomas decomposition,

• Arch. Math. 92(2):111–118, 2009.
• W. Plesken, T. B¨

achler, Counting Polynomials for Linear Codes, Hyperplane Arrangements, and Matroids,

• Doc. Math. 19:285–312, 2014.

JNCF Luminy 2018

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Complexity of quantifier elimination in the theory of ordinary differential equations, in: Lecture Notes Computer Science, 1989, vol. 378, pp. 11–25.

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Complexity of solution of linear systems in rings of differential operators,

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Newton-Puiseux series for non-holonomic D-modules and factoring linear partial differential operators, Moscow Math. J. 9:775–800, 2009.

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Reduction of systems of nonlinear partial differential equations to simplified involutive forms, European J. Appl. Math. 7(6):635–666, 1996.

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JNCF Luminy 2018

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Dimension polynomials of intermediate fields and Krull-type dimension of finitely generated differential field extensions, Mathematics in Computer Science 4(2–3):143–150, 2010.

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A bound for orders in differential Nullstellensatz,

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On the theories of triangular sets,

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JNCF Luminy 2018

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Partial differential equations, Graduate Studies in Mathematics 19, Amer. Math. Soc., 1998.

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JNCF Luminy 2018

References

Wu W.-T., Mathematics mechanization, Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equtaions-solving, Kluwer, 2000.

• D. Wang,

Decomposing polynomial systems into simple systems,

• J. Symbolic Computation 25(3):295–314, 1998.
• Z. Li, D. Wang,

Coherent, regular and simple systems in zero decompositions of partial differential systems,

• Syst. Sci. Math. Sci. 12(Suppl.):43–60, 1999.
• D. Wang,

Elimination practice. Software tools and applications, Imperial College Press, London, 2004.

JNCF Luminy 2018

References

• F. Boulier, D. Lazard, F. Ollivier, M. Petitot,

Representation for the radical of a finitely generated differential ideal, ISSAC 1995, pp. 158–166.

• E. Hubert,

Notes on triangular sets and triangulation-decomposition algorithms.

• I. Polynomial systems. II. Differential systems,

in: F. Winkler, U. Langer (eds.), LNCS, Vol. 2630, Springer, 2003.

• F. Lemaire, M. Moreno Maza, Y. Xie,

The RegularChains library in Maple, SIGSAM Bulletin 39(3):96–97, 2005.

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Comprehensive triangular decomposition, in: Computer Algebra in Scientific Computing, 2007, LNCS, Vol. 4770, Springer.

JNCF Luminy 2018