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Compatible rewriting systems and completion for proving operator - - PowerPoint PPT Presentation

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Compatible rewriting systems and completion for proving operator identities (j.w. Clemens Hofstadler, Clemens G. Raab, and Georg Regensburger) Cyrille Chenavier Johannes Kepler University, Institute for Algebra Journes Nationales du Calcul

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SLIDE 1

Compatible rewriting systems and completion for proving

  • perator identities

(j.w. Clemens Hofstadler, Clemens G. Raab, and Georg Regensburger)

Cyrille Chenavier

Johannes Kepler University, Institute for Algebra Journées Nationales du Calcul Formel 2020 March 2, 2020

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 1 / 14

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SLIDE 2

Formal proofs of operator identities Motivations

Operator identities and membership problems

Objective: formally prove operator identities (e.g., for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . .)

  • perators are terms constructed from basic symbols

"forgetting" the analytic meaning of operators Prove new identities check membership in suitable algebraic structures, e.g.,

  • linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras
  • integro-diff. systems with smooth unknown functions tensor algebras
  • other systems with mixed operations Ore algbras/extensions, tensor rings

Check membership use rewriting theory

  • e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . .
  • simplify a syntactic expression into an equivalent one, e.g.,

∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 2 / 14

slide-3
SLIDE 3

Formal proofs of operator identities Motivations

Operator identities and membership problems

Objective: formally prove operator identities (e.g., for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . .)

  • perators are terms constructed from basic symbols

"forgetting" the analytic meaning of operators Prove new identities check membership in suitable algebraic structures, e.g.,

  • linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras
  • integro-diff. systems with smooth unknown functions tensor algebras
  • other systems with mixed operations Ore algbras/extensions, tensor rings

Check membership use rewriting theory

  • e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . .
  • simplify a syntactic expression into an equivalent one, e.g.,

∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 2 / 14

slide-4
SLIDE 4

Formal proofs of operator identities Motivations

Operator identities and membership problems

Objective: formally prove operator identities (e.g., for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . .)

  • perators are terms constructed from basic symbols

"forgetting" the analytic meaning of operators Prove new identities check membership in suitable algebraic structures, e.g.,

  • linear P.D.E.

’s with constant/polynomial coeff. polynomial/Weyl algebras

  • integro-diff. systems with smooth unknown functions tensor algebras
  • other systems with mixed operations Ore algbras/extensions, tensor rings

Check membership use rewriting theory

  • e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . .
  • simplify a syntactic expression into an equivalent one, e.g.,

∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 2 / 14

slide-5
SLIDE 5

Formal proofs of operator identities Motivations

Operator identities and membership problems

Objective: formally prove operator identities (e.g., for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . .)

  • perators are terms constructed from basic symbols

"forgetting" the analytic meaning of operators Prove new identities check membership in suitable algebraic structures, e.g.,

  • linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras
  • integro-diff. systems with smooth unknown functions tensor algebras
  • other systems with mixed operations Ore algbras/extensions, tensor rings

Check membership use rewriting theory

  • e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . .
  • simplify a syntactic expression into an equivalent one, e.g.,

∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 2 / 14

slide-6
SLIDE 6

Formal proofs of operator identities Motivations

Operator identities and membership problems

Objective: formally prove operator identities (e.g., for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . .)

  • perators are terms constructed from basic symbols

"forgetting" the analytic meaning of operators Prove new identities check membership in suitable algebraic structures, e.g.,

  • linear P.D.E.’s with constant/polynomial coeff. polynomial/Weyl algebras
  • integro-diff. systems with smooth unknown functions tensor algebras
  • other systems with mixed operations Ore algbras/extensions, tensor rings

Check membership use rewriting theory

  • e.g., (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . .
  • simplify a syntactic expression into an equivalent one, e.g.,

∂ ◦ = Id : A ◦ ∂ ◦

  • B

A ◦ B

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 2 / 14

slide-7
SLIDE 7

Formal proofs of operator identities Motivations

Additionnal task

Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g., matrices ⊲ operators may have domains and codomains composition is not defined everywhere e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Solutions

In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility

  • check membership using rew.
  • check "validity" at the end of the calculus

2nd solution: restrict to compatible rew. steps

  • check membership using only "valid" rew. steps
  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 3 / 14

slide-8
SLIDE 8

Formal proofs of operator identities Motivations

Additionnal task

Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g., matrices ⊲ operators may have domains and codomains composition is not defined everywhere e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Solutions

In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility

  • check membership using rew.
  • check "validity" at the end of the calculus

2nd solution: restrict to compatible rew. steps

  • check membership using only "valid" rew. steps
  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 3 / 14

slide-9
SLIDE 9

Formal proofs of operator identities Motivations

Additionnal task

Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g., matrices ⊲ operators may have domains and codomains composition is not defined everywhere e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Solutions

In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility

  • check membership using rew.
  • check "validity" at the end of the calculus

2nd solution: restrict to compatible rew. steps

  • check membership using only "valid" rew. steps
  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 3 / 14

slide-10
SLIDE 10

Formal proofs of operator identities Motivations

Additionnal task

Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g., matrices ⊲ operators may have domains and codomains composition is not defined everywhere e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Solutions

In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility

  • check membership using rew.
  • check "validity" at the end of the calculus

2nd solution: restrict to compatible rew. steps

  • check membership using only "valid" rew. steps
  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 3 / 14

slide-11
SLIDE 11

Formal proofs of operator identities Motivations

Additionnal task

Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g., matrices ⊲ operators may have domains and codomains composition is not defined everywhere e.g., ∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I)

Solutions

In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility

  • check membership using rew.
  • check "validity" at the end of the calculus

2nd solution: restrict to compatible rew. steps

  • check membership using only "valid" rew. steps
  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 3 / 14

slide-12
SLIDE 12

Compatible rew. systems for operator identities Formally prove operator identities

Formal proofs of operator identities in more details

Given: basic operators satisfying identities, e.g., ∂(f ) := f ′,

  • (f ) :=

x

x0

f (t)dt, Eval(f ) := f (x0) are s.t.

  • ∂ = Id − Eval,

∂ ◦ = Id i.e., ∀f :

x

x0 f ′(t)dt = f (x) − f (x0),

x

x0 f (t)dt

′

= f (x) Objective: prove new identities using symbolic methods, e.g., Eval ◦ = 0, i.e., ∀f :

x0

x0 f (t)dt = 0,

follows from

  • − Eval ◦

=

  • ∂ ◦

=

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 4 / 14

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SLIDE 13

Compatible rew. systems for operator identities Formally prove operator identities

Formal proofs of operator identities in more details

Given: basic operators satisfying identities, e.g., ∂(f ) := f ′,

  • (f ) :=

x

x0

f (t)dt, Eval(f ) := f (x0) are s.t.

  • ∂ = Id − Eval,

∂ ◦ = Id i.e., ∀f :

x

x0 f ′(t)dt = f (x) − f (x0),

x

x0 f (t)dt

′

= f (x) Objective: prove new identities using symbolic methods, e.g., Eval ◦ = 0, i.e., ∀f :

x0

x0 f (t)dt = 0,

follows from

  • − Eval ◦

=

  • ∂ ◦

=

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 4 / 14

slide-14
SLIDE 14

Compatible rew. systems for operator identities Formally prove operator identities

Formal proofs of operator identities in more details

Given: basic operators satisfying identities, e.g., ∂(f ) := f ′,

  • (f ) :=

x

x0

f (t)dt, Eval(f ) := f (x0) are s.t.

  • ∂ = Id − Eval,

∂ ◦ = Id i.e., ∀f :

x

x0 f ′(t)dt = f (x) − f (x0),

x

x0 f (t)dt

′

= f (x) Objective: prove new identities using symbolic methods, e.g., Eval ◦ = 0, i.e., ∀f :

x0

x0 f (t)dt = 0,

follows from

  • − Eval ◦

=

  • ∂ ◦

=

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 4 / 14

slide-15
SLIDE 15

Compatible rew. systems for operator identities Noncommutative polynomials and operators

Polynomial description of operators (constant coefficients)

Noncommutative polynomials: polynomial multiplication operator composition, e.g., id − 1 + e, di − 1 ∈ Rd, i, e Indeterminates and generators of the ideal: t

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 and ∂,

  • ,

Eval New identities: belong to the ideal spanned by assumptions, e.g., ei = (id − 1 + e)i − i(di − 1) Remark: membership is not enough if operators have distinct domains and codomains

  • abb−a− − 1 = a(bb− − 1)a− + (aa− − 1): no meaning when

A : U → U, B : V → V are invertible operators

  • for integro-differential operators:

∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 5 / 14

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SLIDE 16

Compatible rew. systems for operator identities Noncommutative polynomials and operators

Polynomial description of operators (constant coefficients)

Noncommutative polynomials: polynomial multiplication operator composition, e.g., id − 1 + e, di − 1 ∈ Rd, i, e Indeterminates and generators of the ideal: t

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 and ∂,

  • ,

Eval New identities: belong to the ideal spanned by assumptions, e.g., ei = (id − 1 + e)i − i(di − 1) Remark: membership is not enough if operators have distinct domains and codomains

  • abb−a− − 1 = a(bb− − 1)a− + (aa− − 1): no meaning when

A : U → U, B : V → V are invertible operators

  • for integro-differential operators:

∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 5 / 14

slide-17
SLIDE 17

Compatible rew. systems for operator identities Noncommutative polynomials and operators

Polynomial description of operators (constant coefficients)

Noncommutative polynomials: polynomial multiplication operator composition, e.g., id − 1 + e, di − 1 ∈ Rd, i, e Indeterminates and generators of the ideal: t

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 and ∂,

  • ,

Eval New identities: belong to the ideal spanned by assumptions, e.g., ei = (id − 1 + e)i − i(di − 1) Remark: membership is not enough if operators have distinct domains and codomains

  • abb−a− − 1 = a(bb− − 1)a− + (aa− − 1): no meaning when

A : U → U, B : V → V are invertible operators

  • for integro-differential operators:

∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 5 / 14

slide-18
SLIDE 18

Compatible rew. systems for operator identities Noncommutative polynomials and operators

Polynomial description of operators (constant coefficients)

Noncommutative polynomials: polynomial multiplication operator composition, e.g., id − 1 + e, di − 1 ∈ Rd, i, e Indeterminates and generators of the ideal: t

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 and ∂,

  • ,

Eval New identities: belong to the ideal spanned by assumptions, e.g., ei = (id − 1 + e)i − i(di − 1) Remark: membership is not enough if operators have distinct domains and codomains

  • abb−a− − 1 = a(bb− − 1)a− + (aa− − 1): no meaning when

A : U → U, B : V → V are invertible operators

  • for integro-differential operators:

∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 5 / 14

slide-19
SLIDE 19

Compatible rew. systems for operator identities Noncommutative polynomials and operators

Polynomial description of operators (constant coefficients)

Noncommutative polynomials: polynomial multiplication operator composition, e.g., id − 1 + e, di − 1 ∈ Rd, i, e Indeterminates and generators of the ideal: t

  • ∂ − Id + Eval = 0,

∂ ◦ − Id = 0 and ∂,

  • ,

Eval New identities: belong to the ideal spanned by assumptions, e.g., ei = (id − 1 + e)i − i(di − 1) Remark: membership is not enough if operators have distinct domains and codomains

  • abb−a− − 1 = a(bb− − 1)a− + (aa− − 1): no meaning when

A : U → U, B : V → V are invertible operators

  • for integro-differential operators:

∂ : Ck+1(I) → Ck(I),

  • : Ck(I) → Ck+1(I),

Eval : Ck+1(I) → Ck+1(I)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 5 / 14

slide-20
SLIDE 20

Compatible rew. systems for operator identities Quiver representations

Quivers represented by operators

  • d

e i C k+1(I) C k(I)

∂ Eval

  • Labelled quivers directed graphs with edges labelled in an alphabet
  • formal link with operators through quiver representations (no injectivity requirement)
  • one letter may be used to label multiple edges (used to represent many identities)

Prove identities ideal membership with a compatible decomposition, e.g., ⊲ ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i ⊲ ∃ parallel paths •

∗

→ • labelled by each monomial Membership may be proved using rewriting 1st method for proving operator identities

Rewrite to zero and then check compatibility

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 6 / 14

slide-21
SLIDE 21

Compatible rew. systems for operator identities Quiver representations

Quivers represented by operators

  • d

e i Ck+1(I) Ck(I)

∂ Eval

  • Labelled quivers directed graphs with edges labelled in an alphabet
  • formal link with operators through quiver representations (no injectivity requirement)
  • one letter may be used to label multiple edges (used to represent many identities)

Prove identities ideal membership with a compatible decomposition, e.g., ⊲ ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i ⊲ ∃ parallel paths •

∗

→ • labelled by each monomial Membership may be proved using rewriting 1st method for proving operator identities

Rewrite to zero and then check compatibility

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 6 / 14

slide-22
SLIDE 22

Compatible rew. systems for operator identities Quiver representations

Quivers represented by operators

  • d

e i C∞(I) C∞(I)

∂ Eval

  • Labelled quivers directed graphs with edges labelled in an alphabet
  • formal link with operators through quiver representations (no injectivity requirement)
  • one letter may be used to label multiple edges (used to represent many identities)

Prove identities ideal membership with a compatible decomposition, e.g., ⊲ ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i ⊲ ∃ parallel paths •

∗

→ • labelled by each monomial Membership may be proved using rewriting 1st method for proving operator identities

Rewrite to zero and then check compatibility

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 6 / 14

slide-23
SLIDE 23

Compatible rew. systems for operator identities Quiver representations

Quivers represented by operators

  • d

e i Ck+1(I) Ck(I)

∂ Eval

  • Labelled quivers directed graphs with edges labelled in an alphabet
  • formal link with operators through quiver representations (no injectivity requirement)
  • one letter may be used to label multiple edges (used to represent many identities)

Prove identities ideal membership with a compatible decomposition, e.g., ⊲ ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i ⊲ ∃ parallel paths •

∗

→ • labelled by each monomial Membership may be proved using rewriting 1st method for proving operator identities

Rewrite to zero and then check compatibility

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 6 / 14

slide-24
SLIDE 24

Compatible rew. systems for operator identities Quiver representations

Quivers represented by operators

  • d

e i e i d Ck+1(I) Ck(I)

∂ Eval

  • Labelled quivers directed graphs with edges labelled in an alphabet
  • formal link with operators through quiver representations (no injectivity requirement)
  • one letter may be used to label multiple edges (used to represent many identities)

Prove identities ideal membership with a compatible decomposition, e.g., ⊲ ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i ⊲ ∃ parallel paths •

∗

→ • labelled by each monomial Membership may be proved using rewriting 1st method for proving operator identities

Rewrite to zero and then check compatibility

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 6 / 14

slide-25
SLIDE 25

Compatible rew. systems for operator identities Quiver representations

Quivers represented by operators

  • d

e i Ck+1(I) Ck(I)

∂ Eval

  • Labelled quivers directed graphs with edges labelled in an alphabet
  • formal link with operators through quiver representations (no injectivity requirement)
  • one letter may be used to label multiple edges (used to represent many identities)

Prove identities ideal membership with a compatible decomposition, e.g., ⊲ ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i ⊲ ∃ parallel paths •

∗

→ • labelled by each monomial Membership may be proved using rewriting 1st method for proving operator identities

Rewrite to zero and then check compatibility

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 6 / 14

slide-26
SLIDE 26

Compatible rew. systems for operator identities Quiver representations

Quivers represented by operators

  • d

e i Ck+1(I) Ck(I)

∂ Eval

  • Labelled quivers directed graphs with edges labelled in an alphabet
  • formal link with operators through quiver representations (no injectivity requirement)
  • one letter may be used to label multiple edges (used to represent many identities)

Prove identities ideal membership with a compatible decomposition, e.g., ⊲ ei = (id − 1 + e)i − i(di − 1) = idi − i + ei − idi + i ⊲ ∃ parallel paths •

∗

→ • labelled by each monomial Membership may be proved using rewriting 1st method for proving operator identities

Rewrite to zero and then check compatibility

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 6 / 14

slide-27
SLIDE 27

Compatible rew. systems for operator identities Provide compatible decompositions

Illustrating example

Consider the inhomogeneous linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (1) Assumption: (1) can be factored into the 1st order equations y′(x) − B2(x)y(x) = z(x) and z′(x) − B1(x)z(x) = r(x) General solution given by y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt (2) where Hi(x) is s.t. H′

i (x) − Bi(x)Hi(x) = 0 and Hi(x)−1 exists

Illustration of the rewriting/quiver approach: formally prove that (2) is a solution of (1)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 7 / 14

slide-28
SLIDE 28

Compatible rew. systems for operator identities Provide compatible decompositions

Illustrating example

Consider the inhomogeneous linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (1) Assumption: (1) can be factored into the 1st order equations y′(x) − B2(x)y(x) = z(x) and z′(x) − B1(x)z(x) = r(x) General solution given by y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt (2) where Hi(x) is s.t. H′

i (x) − Bi(x)Hi(x) = 0 and Hi(x)−1 exists

Illustration of the rewriting/quiver approach: formally prove that (2) is a solution of (1)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 7 / 14

slide-29
SLIDE 29

Compatible rew. systems for operator identities Provide compatible decompositions

Illustrating example

Assumptions: prove that y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is solution of

  • (∂ − B1) ◦ (∂ − B2)

(y(x)) = r(x) where Hi is s.t. H′

i (x) − Bi(x)Hi(x) = 0 and Hi(x)−1 exists

Algebraic part: X := {h1, h2, b1, b2, ˜ h1, ˜ h2, i, d}, I := I(f1, . . . f5) ⊂ RX, where f1 := dh1 − h1d − b1h1, f2 := dh2 − h2d − b2h2, f3 := h1˜ h1 − 1, f4 := h2˜ h2 − 1, f5 := di − 1 Claim: f admits a compatible decomposition, where f := (d − b1)(d − b2)h2i˜ h2h1i˜ h1 − 1

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 8 / 14

slide-30
SLIDE 30

Compatible rew. systems for operator identities Provide compatible decompositions

First method

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • Fact: F := {f1, . . . , f5}

⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) is a compatible decomposition of f the claim is proven

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 9 / 14

slide-31
SLIDE 31

Compatible rew. systems for operator identities Provide compatible decompositions

First method

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) is a compatible decomposition of f the claim is proven

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 9 / 14

slide-32
SLIDE 32

Compatible rew. systems for operator identities Provide compatible decompositions

First method

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) is a compatible decomposition of f the claim is proven

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 9 / 14

slide-33
SLIDE 33

Compatible rew. systems for operator identities Provide compatible decompositions

First method

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) is a compatible decomposition of f the claim is proven

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 9 / 14

slide-34
SLIDE 34

Compatible rew. systems for operator identities Provide compatible decompositions

First method

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) is a compatible decomposition of f the claim is proven

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 9 / 14

slide-35
SLIDE 35

Compatible rew. systems for operator identities Provide compatible decompositions

First method

Represented quiver: we need 2nd order derivative/integration and regularity assumptions

  • d

d i i b1 b2 h2 h1 ˜ h1 h2 ˜ h2 h1 Fact: F := {f1, . . . , f5} ⇒ f

∗

→F 0 using an orientation of fi’s dh1 h1d + b1h1, dh2 h2d + b2h2, h1˜ h1 1, h2˜ h2 1, di 1 and keeping track of cofactors, we get f = f1i˜ h1 + (d − b1)f2i˜ h2h1i˜ h1 + f3 + (d − b1)f4h1i˜ h1 +(d − b1)h2f5˜ h2h1i˜ h1 + h1f5˜ h1 (3) By a case analysis: (3) is a compatible decomposition of f the claim is proven

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 9 / 14

slide-36
SLIDE 36

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Second method

Objective: restrict to rew. steps s.t.

We only use "valid" computations

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible the first method involved invalid computations ⊲ h2d → dh2 − b2h2 is compatible

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 10 / 14

slide-37
SLIDE 37

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Second method

Objective: restrict to rew. steps s.t.

We only use "valid" computations

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible the first method involved invalid computations ⊲ h2d → dh2 − b2h2 is compatible

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 10 / 14

slide-38
SLIDE 38

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Second method

Objective: restrict to rew. steps s.t.

We only use "valid" computations

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible the first method involved invalid computations ⊲ h2d → dh2 − b2h2 is compatible

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 10 / 14

slide-39
SLIDE 39

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Second method

Objective: restrict to rew. steps s.t.

We only use "valid" computations

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible the first method involved invalid computations ⊲ h2d → dh2 − b2h2 is compatible

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 10 / 14

slide-40
SLIDE 40

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Second method

Objective: restrict to rew. steps s.t.

We only use "valid" computations

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible the first method involved invalid computations ⊲ h2d → dh2 − b2h2 is compatible

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 10 / 14

slide-41
SLIDE 41

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Second method

Objective: restrict to rew. steps s.t.

We only use "valid" computations

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible the first method involved invalid computations ⊲ h2d → dh2 − b2h2 is compatible

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 10 / 14

slide-42
SLIDE 42

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Second method

Objective: restrict to rew. steps s.t.

We only use "valid" computations

Signatures: σ(dh2) = {•

∗

→ •,

  • ∗

→ •}, σ(h2d) = {•

∗

→ •}, σ(b2h2) = {•

∗

→ •}

  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2 Definition: a rew. rule m → g is Q-compatible if σ(m) ⊆ σ(g), e.g., ⊲ dh2 → h2d + b2h2 is not compatible the first method involved invalid computations ⊲ h2d → dh2 − b2h2 is compatible

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 10 / 14

slide-43
SLIDE 43

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Theorem Let Q be a quiver labelled by X, let F ⊂ RX and let g ∈ RX. Assume that each rew. rule is Q-compatible and g

∗

→F 0.

  • The obtained decomposition of g is compatible
  • For all representations of Q s.t. all elements of F are mapped to 0, g is also mapped

to 0 Remark

Using the Theorem:

  • representation(s) of the quiver map any polynomial to the operator(s) it represents
  • elements of F polynomial expressions of known operator identities
  • g polynomial expression of the identity we wish to prove
  • g

∗

→F 0 with compatible rew. rules only the identity is proven

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 11 / 14

slide-44
SLIDE 44

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-45
SLIDE 45

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-46
SLIDE 46

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-47
SLIDE 47

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-48
SLIDE 48

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-49
SLIDE 49

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-50
SLIDE 50

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-51
SLIDE 51

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Completion

Motivating example: consider as previously F := {f1, . . . , f5} and f

  • for the deglex order s.t. b1, h1 < d < b2 < h2, we get the compatible rew. rules

dh1 h1d + b1h1, h2d dh2 − b2h2, h1˜ h1 1, h2˜ h2 1, di 1

  • problem: f does not rewrite into 0
  • we need a compatible completion procedure

Adaptation of the Buchberger’s proc.: the compatible monomial h2di induces SP = dh2i − b2h2i − h2, LM(SP) = b2h2i ⊲ σ(b2h2i) = {•

∗

→ •} ⊆ σ(dh2i) ∩ σ(h2)

  • we keep f6 := dh2i − b2h2i − h2
  • d

d b1 b2 i i h1 h1 h2 h2 ˜ h1 ˜ h2

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 12 / 14

slide-52
SLIDE 52

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ G : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where

  • A0, A1 are functions of class Ck
  • r is a function of class Ck

If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a function of classe Ck+2 solution of (4)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 13 / 14

slide-53
SLIDE 53

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ G : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where

  • A0, A1 are functions of class Ck
  • r is a function of class Ck

If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a function of classe Ck+2 solution of (4)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 13 / 14

slide-54
SLIDE 54

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ G : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where

  • A0, A1 are analytic functions
  • r is an analytic function

If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is an analytic function solution of (4)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 13 / 14

slide-55
SLIDE 55

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ G : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where

  • A0, A1 are n × n matrices of functions of class Ck
  • r is a vector of n functions of class Ck

If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a vector of n functions of class Ck+2 solution of (4)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 13 / 14

slide-56
SLIDE 56

Compatible rew. systems for operator identities Provide compatible decompositions: second method

Many proofs at once

Using completion: letting G := F ∪ {f6}, we have f

∗

→G 0 From the compatibility theorem: for all representations ϕ of Q ∀g ∈ G : ϕ(g) = 0 ⇒ ϕ(f ) = 0 Consequences: let us consider the linear O.D.E. y′′(x) + A1(x)y′(x) + A0(x)y(x) = r(x) (4) where

  • A0, A1 are · · ·
  • r is a · · ·

If (4) may be factored into 1st order O.D.E.’s with homogeneous invertible sol. Hi, y(x) = H2(x)

x

x2

H2(t)−1H1(t)

t

x1

H1(u)−1r(u) du dt is a · · · solution of (4)

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 13 / 14

slide-57
SLIDE 57

Compatible rew. systems for operator identities Conclusion

Summary

Compatible rewriting rew. rules respects parallel paths of a labelled quiver

  • used to prove operator identities without checking each computational step
  • existence of a (partial) completion procedure
  • implementation Mathematica package OperatorGB

References: t

  • arXiv ref: arXiv:2002.03626
  • package link: http://gregensburger.com/softw/OperatorGB/

THANK YOU FOR YOUR ATTENTION!

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 14 / 14

slide-58
SLIDE 58

Compatible rew. systems for operator identities Conclusion

Summary

Compatible rewriting rew. rules respects parallel paths of a labelled quiver

  • used to prove operator identities without checking each computational step
  • existence of a (partial) completion procedure
  • implementation Mathematica package OperatorGB

References: t

  • arXiv ref: arXiv:2002.03626
  • package link: http://gregensburger.com/softw/OperatorGB/

THANK YOU FOR YOUR ATTENTION!

  • C. Chenavier

JKU, Institute for Algebra March 2, 2020 14 / 14