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Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Tutorial: Numerical Algebraic Geometry

Back to classical algebraic geometry... with more computational power and hybrid symbolic-numerical algorithms Anton Leykin

Georgia Tech

Waterloo, November 2011

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Outline

Homotopy continuation predictor-corrector numerical methods, Newton’s method, (global) homotopy continuation scenarios Singular isolated solutions regularization of singular solutions, deflation, dual spaces/inverse systems Positive dimension witness sets, numerical irreducible decomposition, numerical primary decomposition Certified homotopy tracking numerical zeros, α-theory of Smale, heuristic vs. rigorous path-tracking

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Computational algebraic geometry

What is the game?

• Level 0: Given a system of polynomial equations in K[x1, ..., xn]

with finitely many solutions, SOLVE. ( K could be Q, Z/pZ, R, C, ... )

• Level 1+: Describe positive-dimensional solutions (curves,

surfaces, ...) Classical methods “generalize” linear algebra:

• Gröbner basis: a generalization of Gaussian reduction;
• Resultant: a generalization of determinant.

These methods are symbolic.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Linear Algebra

• Numerical Linear Algebra

  

• 

 

• Algebraic Geometry
• Numerical Algebraic Geometry

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Applications

Robotics: Stewart-Gough platforms. Griffis-Duffy platform: the solution contains a curve of degree 28. ℓ1 ℓ2 ℓ3 ℓ4 s1 s2 p✑✑ ✑ ✸ Enumerative algebraic geometry: solutions of Schubert problems. ... control theory, optimization, computer vision, math biology, real algebraic geometry, algebraic curves ...

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Polynomial homotopy continuation

• Target system: n equations in n variables,

F(x) = (f1(x), . . . , fn(x)) = 0, where fi ∈ R = C[x] = C[x1, ..., xn] for i = 1, ..., n.

• Start system: n equations in n variables:

G(x) = (g1(x), . . . , gn(x)) = 0, such that it is easy to solve.

• Homotopy: for γ ∈ C \ {0} consider

H(x, t) = (1 − t)G(x) + γtF(x), t ∈ [0, 1].

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Example

target start f1 = x4

1x2 + 5x2 1x3 2 + x3 1 − 4

g1 = x5

1 − 1

f2 = x2

1 − x1x2 + x2 − 8

g2 = x2

2 − 1

Start solutions → target solutions: H(x, t) = 0 implies dx dt = − ∂H ∂x −1 ∂H ∂t .

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Example

target start f = −x2 + 2 g = x2 − 1 The solution of the homotopy equation H(x, t) = (1 − t)g(x) + tf(x) = (1 − 2t)x2 − 1 + 3t = 0 is singular for t = 1/3.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Randomization

• Note: the complement of a complex algebraic variety is

connected. space of polynomials

f g

• For all but finite number of γ ∈ C the homotopy

H(x, t) = (1 − t)G(x) + γtF(x). is regular for 0 ≤ t < 1.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Global picture

Optimal homotopy:

• the continuation paths are regular;
• the homotopy establishes a bijection

between the start and target solutions. Possible singular scenarios: non-generic diverging paths multiple solutions

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Numerical algebraic geometry

• Sommese, Verschelde, and Wampler, Introduction to Numerical

AG (2005)

• Sommese and Wampler, The numerical solution of systems of

polynomials (2005) Software:

• PHCpack (Verschelde);
• HOM4PS (group of T.Y.Li);
• Bertini (group of Sommese);
• NAG4M2: Numerical Algebraic Geometry for Macaulay2 (L.).

and more, e.g.: Maple’s ROOTFINDING[HOMOTOPY].

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Possible improvements

• Parallel computation:

Paths are mutually independent ⇒ linear speedups.

• Minimize the number of diverging paths:
• Total degree: Number of start solutions =

product of degrees of equations (Bézout bound).

• Polyhedral homotopies: Number of start

solutions = mixed volume of sparse system (BKK bound).

• Optimal homotopies:
• Cheater’s homotopy;
• Special homotopies: e.g., Pieri homotopy.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Multiple solutions

In general, with probability 1, the picture looks like this: Singular end games [Morgan, Sommese, Wampler (1991)]:

• power-series method;
• Cauchy integral method;
• trace method.

Deflation:

• regularizes an isolated singular solution;
• restores quadratic convergence of the Newton’s method.

How to describe a singularity?

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Cauchy integral endgame

• An implication of Cauchy residue theorem:

Let y : U → C holomorphic on a simply connected U ⊂ C, a ∈ C, and C ⊃ C ≃ S1 be a contour winding I(C, a) times around a. Then y(a) = 1 2πiI(C, a)

• C

y(z) z − a dt,

• H(x, t) = 0 defines (a possibly multivalued function) x = x(t) in

a neighborhood of t = 1.

• Idea: as the homotopy tracker approaches a singular x∗ = x(1)

use Cauchy integral to compute x∗ staying away from x∗.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Winding numbers

(1 2 3 4 5)(6 7 8)(9 10)

|1 − t| = ε

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Cauchy integral endgame

• 1. Pick a point on ˜

x = x(˜ t), a solution to H(x, t) = 0 for t = ˜ t ∈ R; let ε = 1 − ˜ t.

• 2. Track the path

C = {x(1 − εeiθ ˜

I) | θ ∈ [0, 2π]},

where ˜ I > 0 is such that ˜ x = x(1 − εeiθ ˜

I) ⇒ θ ∈ {0, 2π}.

• 3. Let y(z) = x(1 − z ˜

I), then y(z) is holonomic for |z| < ε (if ε ≪ 1).

• 4. Find numerically the integral

x(1) = y(0) = 1 2π

• |z|=ε

y(z) z dz = 1 2π

• [0,2π]

x(1 − εeiθ ˜

I) dθ.

(Note: one may use samples made when tracking the path C.)

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Newton’s method: x(n+1) = x(n) − f(x(n)) f ′(x(n))

Example 1: f(x) = x(x − 1)3, x(0) = 0.1

x(1) = −0.05000000000000000000000000000000000000000000000000 x(2) = −0.00625000000000000000000000000000000000000000000000 x(3) = −0.00011432926829268292682926829268292682926829268293 x(4) = −0.00000003919561993882928315798471103711494222972094 x(5) = −0.00000000000000460888914457438597268761599543603706 x(6) = −0.00000000000000000000000000006372557744092567103642

Example 2: f(x) = x2(x − 1)3, x(0) = 0.1

x(1) = 0.04000000000000000000000000000000000000000000000000 x(2) = 0.01866666666666666666666666666666666666666666666667 x(3) = 0.00905920745920745920745920745920745920745920745921 x(4) = 0.00446662546689373374865785737653016156369492056043 x(5) = 0.00221818070337351048684295922675846246257988477728 x(6) = 0.00110537952927547542499858913840929687677679537995

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Deflation method

Let f(x) = (f1(x), ..., fN(x)), N ≥ n, fi(x) ∈ C[x] = C[x1, ..., xn]. Let A(x) = ∂fi ∂xj

• ∈ CN×n be the Jacobian matrix.

Given: an approximation x(0) of an exact isolated solution x∗, which is singular, i.e., corank A(x∗) = n − rank A(x∗) > 0. Newton’s method in homotopy continuation loses quadratic convergence around x∗. Is there a way to restore the convergence?

• Want: a symbolic procedure that “makes” x∗ regular.
• Rules:
• New variables are allowed.
• Assume that the numerical rank of A(x(0)) equals A(x∗).

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Deflation step: create an augmented system in C[x, a]

• 1. Introduce n new variables a;
• 2. Add equations coming from A(x)a = 0;
• Example. Let f1 = x3

1 + x1x2 2, f2 = x1x2 2 + x3 2, f3 = x2 1x2 + x1x2 2 and

x∗ = 0.    ∂1 ∂2 f1 3x2

1 + x2 2

2x1x2 f2 x2

2

2x1x2 + 3x2

2

f3 2x1x2 + x2

2

x2

1 + 2x1x2

  

• a1

a2

• = 0.
• 3. Compute the rank r of A(x∗); (r = 0 for our example)
• 4. Add n − r random linear equations.
• 5. Find the solution (x∗, a∗) of the augmented system; (8 equations)
• 6. Repeat if (x∗, a∗) is singular. (2 steps for the example)

Theorem (L., Verschelde, Zhao)

The multiplicity of (x∗, a∗) in the augmented system is smaller than that of x∗ in the original system.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Multiplicity

Staircases for I = f1, f2, f3, where f1 = x3

1 + x1x2 2, f2 = x1x2 2 + x3 2, f3 = x2 1x2 + x1x2 2.

✻ ✲ ❤ ❤ ❤ ❤ ❤ ❤ ① ① ①

x1x2

2 + x3 2

x2

1x2 + x1x2 2

x3

1 + x1x2 2

① ❤

x4

2

ω = (−1, −2)

☛ ✻ ✲ ❤ ❤ ❤ ❤ ❤ ❤ ① ① ①

x2

1x2 + x1x2 2

x1x2

2 + x3 1

x3

2 + x1x2 2

① ❤

x4

1

ω = (−2, −1)

✙ Multiplicity is the number of integer points under the staircase.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Dual space (local inverse system)

• For x ∈ Cn, let ∆β

x : R → C be a linear functional,

∆β

x(f) = (∂β · f)(x) = ∂|β|f

∂β (x), f ∈ R.

• For an ideal I, the dual space Dx[I] is the subspace of

SpanC{∆β

x} of the functionals that annihilate I.

• Filter by order:

D(0)

x [I] ⊂ D(1) x [I] ⊂ D(2) x [I] ⊂ . . .

where D(d)

x [I] is the set of functionals of order at most d.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Macaulay array

• For I = f1, . . . , fm, the deflation matrix A(d)

I (x) is a part of the

Macaulay array, the infinite matrix with entries ∂ ∂xβ (xαfi)

• ((i,α),β)

where |α| < d and |β| ≤ d.

• For example, for I = f1, f2 ⊂ C[x, y],

A(2)

I

=          id ∂x ∂y ∂2

x

∂x∂y ∂2

y

f1 ∗ ∗ ∗ ∗ ∗ ∗ f2 ∗ ∗ ∗ ∗ ∗ ∗ xf1 ∗ ∗ ∗

∂ ∂x2 (xf1)

∗ ∗ xf2 ∗ ∗ ∗ ∗ ∗ ∗ yf1 ∗ ∗ ∗ ∗ ∗ ∗ yf2 ∗ ∗ ∗ ∗ ∗ ∗         

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Dual spaces and deflation

• For x ∈ V (I), get D(d)

x [I] by computing ker A(d) I (x).

• For the running example,

D0[I] = Span{ ∆(3,0) − ∆(2,1) − ∆(1,2) + ∆(0,3), ∆(2,0), ∆(1,1), ∆(0,2), ∆(1,0), ∆(0,1), ∆(0,0) }. The leading terms with respect to ω = (2, 1) correspond to the monomials under the staircase for the standard basis for ω = (−2, −1). ✻ ✲ ❤ ❤ ❤ ❤ ❤ ❤ ① ① ①

x2

1x2 + x1x2 2

x1x2

2 + x3 1

x3

2 + x1x2 2

① ❤

x4

1

ω = (−2, −1)

✙

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Deflation continued...

Idea of proof of termination for deflation:

• Deflation "deflates the staircase";
• the multiplicity of becomes 1 after a finite number of steps.

Related work:

• Dual spaces: Macaulay (1916), ..., Stetter (1993), Mourrain

(1997), Dayton and Zeng (2005), Krone (2011).

• Deflation: Lyapunov (1900?) , ..., Ojika et al (1987), Lecerf

(2002), L. et al (2006), ..., Lihong Zhi et al (2010).

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Higher-dimensional solution sets

• Let I = (f1, . . . , fN) be an ideal of C[x1, . . . , xn].
• Goal: Understand the variety

X = V(I) = {x ∈ Cn | ∀f ∈ I, f(x) = 0}.

• A witness set for an equidimensional component Y of X:
• a generic “slicing” plane L with dim L = codim Y
• witness points wY,L = Y ∩ L
• (generators of I)

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Numerical irreducible decomposition

• Homotopy mapping wY,L → wY,L′:

HL,L′,γ(x, t) =

• f(x)

(1 − t)L(x) + γtL′(x)

• , t ∈ [0, 1].

L L’

• Monodromy action: a permutation on wY,L

is produced by homotopy HL,L′,γ followed by HL′,L,γ′ for random γ, γ′ ∈ C.

• Irreducible decomposition: a partition of the

witness set wY,L stable under this action.

• Linear trace test: the average of the

points in a witness set of an irreducible component behaves linearly.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Example with an embedded component

Example

I = (f1, f2), where f1 = x2(y + 1) and f2 = xy(y + 1). y + 1 = 0 x = 0

• ✠

solution set of x · x(y + 1) = 0; y · x(y + 1) = 0. Numerical irreducible decomposition* sees two 1-dimensional components ...

*NID reference: Sommese, Verschelde, Wampler “Numerical decomposition

• f the solution sets...” (2001)

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Example with an embedded component

Example

I = (f1, f2), where f1 = x2(y + 1) and f2 = xy(y + 1). y + 1 = 0 x = 0 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

• ✠

solution set of x · x(y + 1) = 0; y · x(y + 1) = 0. Numerical irreducible decomposition* sees two 1-dimensional components ...

*NID reference: Sommese, Verschelde, Wampler “Numerical decomposition

• f the solution sets...” (2001)

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Example with an embedded component

Example

I = (f1, f2), where f1 = x2(y + 1) and f2 = xy(y + 1). y + 1 = 0 x = 0 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✉(0, 0)

• ✠

solution set of x · x(y + 1) = 0; y · x(y + 1) = 0. Numerical irreducible decomposition* sees two 1-dimensional components ... ... but does not discover the embedded point.

*NID reference: Sommese, Verschelde, Wampler “Numerical decomposition

• f the solution sets...” (2001)

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Deflation matrix, system, and ideal

Example (Deflation ideal of order d = 2)

• riginal system f(x, y)

→ deflation matrix A(2)

I (x, y)

         ∂x ∂y ∂2

x

∂x∂y ∂2

y

f1 ∗ ∗ ∗ ∗ ∗ f2 ∗ ∗ ∗ ∗ ∗ xf1 ∗ ∗ 6x(y + 1) ∗ ∗ xf2 ∗ ∗ ∗ ∗ ∗ yf1 ∗ ∗ ∗ ∗ ∗ yf2 ∗ ∗ ∗ ∗ ∗          ✻

∂2(xf1) ∂x2

Deflation ideal

I(2) = f, D(2)f ⊂ C[x, y, a]

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Deflation matrix, system, and ideal

Example (Deflation ideal of order d = 2)

• riginal system f(x, y)

→ deflation matrix A(2)

I (x, y)

→ → deflation system D(2)f(x, y, a)          ∂x ∂y ∂2

x

∂x∂y ∂2

y

f1 ∗ ∗ ∗ ∗ ∗ f2 ∗ ∗ ∗ ∗ ∗ xf1 ∗ ∗ 6x(y + 1) ∗ ∗ xf2 ∗ ∗ ∗ ∗ ∗ yf1 ∗ ∗ ∗ ∗ ∗ yf2 ∗ ∗ ∗ ∗ ∗                 ax ay axx axy ayy        =: D(2)f(x, y, a) ✻

∂2(xf1) ∂x2

✻ C[x, y, a]6

Deflation ideal

I(2) = f, D(2)f ⊂ C[x, y, a]

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Deflation matrix, system, and ideal

Example (Deflation ideal of order d = 2)

• riginal system f(x, y)

→ deflation matrix A(2)

I (x, y)

→ → deflation system D(2)f(x, y, a) → → deflation ideal I(2)          ∂x ∂y ∂2

x

∂x∂y ∂2

y

f1 ∗ ∗ ∗ ∗ ∗ f2 ∗ ∗ ∗ ∗ ∗ xf1 ∗ ∗ 6x(y + 1) ∗ ∗ xf2 ∗ ∗ ∗ ∗ ∗ yf1 ∗ ∗ ∗ ∗ ∗ yf2 ∗ ∗ ∗ ∗ ∗                 ax ay axx axy ayy        =: D(2)f(x, y, a) ✻

∂2(xf1) ∂x2

✻ C[x, y, a]6

Deflation ideal

I(2) = f, D(2)f ⊂ C[x, y, a]

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Deflated variety

• For the given variety X = V (I) = {x | f(x) = 0 for all f ∈ I}

define deflated variety of order d: X(d) = V (I(d)) ⊂ CB(n,d), where B(n, d) = n − 1 + n+d+1

d

• .
• Projection: πd : CB(n,d) → Cn,

πd(x, a) → x; πdX(d) = X.

• A component Y ⊂ X is called visible at order d if Y = πdZ for an

isolated component Z ⊂ X(d). s(0, 0)

Theorem (L.)

Every component is visible at some order d.

Example

For the running example d = 1 is sufficient.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Example: I = (f1, f2), f1 = x2(y + 1), f2 = xy(y + 1)

• Isolated components of X = V (I):

V (y + 1), V (x).

• Additional equations A(1)

I a = 0:

• 2x(y + 1)

x2 y(y + 1) x(2y + 1) ax ay

• = 0.
• Isolated components of X(1) = V (I(1)):

V (y + 1, ay), V (x, y), V (x, ax), V (x, y + 1)

• The first three project onto the components of X, the last one (a

“pseudo-component”) projects onto a singular point.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Witness sets

For an irreducible subvariety Y ⊂ X = V (I), where I ⊂ R is an ideal, a witness set consists of

• a generic “slicing” plane L with dim L = codim Y
• witness points w = Y

∩ L

• generators of I

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Witness sets

For an irreducible subvariety Y ⊂ X = V (I), where I ⊂ R is an ideal, a generalized witness set consists of

• a generic “slicing” plane L with dim L = codim Y (d)
• witness points w = Y (d) ∩ L and their projections via πd
• generators of I(d)

Definition: Y (d) is an isolated irreducible component of the deflated variety X(d) mapping onto Y under projection πd.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

The algorithm and NPD concept

• Idea of the algorithm: Use numerical irreducible decomposition
• f a deflated variety to find (generalized) witness sets

representing components.

• Definition: Numerical primary decomposition is a collection of

such witness sets, one per component.

• Deficiencies:
• There is an apriori bound on the order d of needed to make all

components visible, however it is not practical.

• Pseudocomponents (projections of components of X(d) that are

not components of X) are hard to eliminate.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Local global?

Local knowledge:

• Dx[I] describes the local ring (R/I)x = Rx/Ix.
• A generic point on a component Y is a smooth point of Y that

does not belong to any component not containing Y properly.

• A generic point x ∈ Y together with the algorithm for computing

D(d)

x [I] describe Y .

Global knowledge:

• Given a NPD (as a collection of witness sets), mark one generic

point on each component.

• The set of marked points describe the ideal I.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Path switching

Applications in mathematics where path certification is desirable:

• Numerical irreducible

decomposition algorithms

• Galois group computation based
• n monodromy
• Problems where the root count is

impossible by other methods

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Newton’s method and approximate zeros

Given f ∈ C[x], consider the Newton operator associated to f, N(f)(x) = x − Df(x)−1f(x), where Df(x) is the n × n derivative (Jacobian) matrix of f at x ∈ Cn.

• Definition: x ∈ Cn is an approximate zero of f with associated

zero η ∈ Cn if N(f)l(x) − η ≤ x − η 22l−1 , l ≥ 0.

• γ-theorem(Smale): Let x ∈ Cn, η ∈ f −1(0), and

x − η ≤ 3 − √ 7 2γ(f, x) , where γ(f, x) = sup

k≥2

• Df(x)−1Dkf(x)

k!

• 1

k−1

. Then x is an approximate zero of f associated to η.

• Hauenstein, Sottile: alphaCertify, software for certification of

regular solutions.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Smale’s α theory

• α-theorem: Let

β(f, x) = x − N(f)(x) = Df(x)−1f(x) . Then α(f, x) = β(f, x)γ(f, x) < 0.15767 certifies that x is an approximate zero of f.

• "robust" theorem: Let x ∈ Cn with α(f, x) < 0.03. If

x − y < 1 20γ(f, x) , then y is an approximate zero associated to the same zero as x.

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Newton’s operator attraction basins

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Certified regions

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Robust regions

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Linear and segment homotopy

• Let H(d) be the space of systems of homogeneous polynomials
• f fixed degrees (d) = (d1, . . . , dn) (with the Bombieri-Weyl

norm).

• Consider f, g ∈ S = {f ∈ H(d) : f = 1} ⊂ H(d).
• Using α-theory we design certified homotopy tracking (CHT)

algorithm that tracks a linear homotopy on S (assuming BSS model of computation).

• The “robust” α-theory leads to the robust CHT algorithm (Beltrán,

L.):

g f

Take input f, g with coefficients in Q[i]. Use the segment homotopy: t → ht = (1 − t)g + tf, t = [0, 1]. All computations use exact linear algebra over Q[i].

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Future

• General methods
• (Numerical) local ring structure
• (Numerical) primary decomposition
• Real solutions, real homotopy continuation
• Certification
• Generalizations to higher order methods
• Certification of singular isolated solutions
• Certification of (irreducible/primary) decomposition
• Upcoming events
• SI(AG)2: SIAM activity group in algebraic geometry
• IMA PI summer program for graduate students on

Algebraic Geometry for Applications June 18th – July 6th at Georgia Tech