Fast polynomial reduction for generic bivariate ideals Joris van - PowerPoint PPT Presentation

Fast polynomial reduction for generic bivariate ideals Joris van der Hoeven, Robin Larrieu Laboratoire d’Informatique de l’Ecole Polytechnique (LIX) CARAMBA Seminar – Nancy 23 / 05 / 2019 Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 1 / 27

Introduction Let � A , B � be the ideal generated by A and B ( A , B ∈ K [ X , Y ]). Given P ∈ K [ X , Y ], check if P ∈ � A , B � . (ideal membership test) Compute a normal form of ¯ P ∈ K [ X , Y ] / � A , B � . (computation in the quotient algebra) Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 2 / 27

Introduction Let � A , B � be the ideal generated by A and B ( A , B ∈ K [ X , Y ]). Given P ∈ K [ X , Y ], check if P ∈ � A , B � . (ideal membership test) Compute a normal form of ¯ P ∈ K [ X , Y ] / � A , B � . (computation in the quotient algebra) Classical solution using Gr¨ obner bases . Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 2 / 27

Introduction Let � A , B � be the ideal generated by A and B ( A , B ∈ K [ X , Y ]). Given P ∈ K [ X , Y ], check if P ∈ � A , B � . (ideal membership test) Compute a normal form of ¯ P ∈ K [ X , Y ] / � A , B � . (computation in the quotient algebra) Classical solution using Gr¨ obner bases . Fast Gr¨ obner basis algorithms rely on linear algebra (ex: F4, F5. . . ) Can we do it with polynomial arithmetic? Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 2 / 27

Introduction Let � A , B � be the ideal generated by A and B ( A , B ∈ K [ X , Y ]). Given P ∈ K [ X , Y ], check if P ∈ � A , B � . (ideal membership test) Compute a normal form of ¯ P ∈ K [ X , Y ] / � A , B � . (computation in the quotient algebra) Classical solution using Gr¨ obner bases . Fast Gr¨ obner basis algorithms rely on linear algebra (ex: F4, F5. . . ) Can we do it with polynomial arithmetic? Given a Gr¨ obner basis G , can we reduce P modulo G faster? Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 2 / 27

Introduction Let � A , B � be the ideal generated by A and B ( A , B ∈ K [ X , Y ]). Given P ∈ K [ X , Y ], check if P ∈ � A , B � . (ideal membership test) Compute a normal form of ¯ P ∈ K [ X , Y ] / � A , B � . (computation in the quotient algebra) Classical solution using Gr¨ obner bases . Fast Gr¨ obner basis algorithms rely on linear algebra (ex: F4, F5. . . ) Can we do it with polynomial arithmetic? Given a Gr¨ obner basis G , can we reduce P modulo G faster? Are these ideas useful to compute G faster? Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 2 / 27

Introduction Main result For generic ideals in two variables, reduction is possible with quasi-optimal complexity. If A,B are given in total degree and if we use the degree-lexicographic order, then the Gr¨ obner basis can also be computed efficiently. References van der Hoeven, L. Fast reduction of bivariate polynomials obner bases (ISSAC ’18). with respect to sufficiently regular Gr¨ van der Hoeven, L. Fast Gr¨ obner basis computation and polynomial reduction for generic bivariate ideals (to appear in AAECC). Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 3 / 27

Outline Key ingredients 1 Dichotomic selection strategy Truncated basis elements Rewriting the equation Vanilla Gr¨ obner bases 2 Definition Terse representation Reduction algorithm Case of the grevlex order 3 Presentation of the setting Concise representation Reduction algorithm Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 4 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Outline Key ingredients 1 Dichotomic selection strategy Truncated basis elements Rewriting the equation Vanilla Gr¨ obner bases 2 Case of the grevlex order 3 Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 5 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Presentation of the problem Y lead. monom. of G K -basis of K [ X , Y ] / I X A , B : O ( n 2 ) coefficients K [ X , Y ] / I : dimension O ( n 2 ) G : O ( n 3 ) coefficients ( O ( n 2 ) for each G i ) Reduction using G needs at least O ( n 3 ) = ⇒ reduction with less information? Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 6 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Presentation of the problem Theorem (van der Hoeven – ACA 2015) The extended reduction of P modulo G can be computed in quasi-linear time for the size of the equation � P = Q i G i + R i Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 7 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Presentation of the problem Theorem (van der Hoeven – ACA 2015) The extended reduction of P modulo G can be computed in quasi-linear time for the size of the equation � P = Q i G i + R i But this equation has size Θ( n 3 ) and we would like to achieve ˜ O ( n 2 ) complexity. . . Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 7 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Presentation of the problem Theorem (van der Hoeven – ACA 2015) The extended reduction of P modulo G can be computed in quasi-linear time for the size of the equation � P = Q i G i + R i But this equation has size Θ( n 3 ) and we would like to achieve ˜ O ( n 2 ) complexity. . . = ⇒ Somehow reduce the size of the equation. Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 7 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Dichotomic selection strategy The extended reduction is not unique: several ways to reduce each term. The remainder is unique if G is a Gr¨ obner basis. The quotients depend on a selection strategy . Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 8 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Dichotomic selection strategy The extended reduction is not unique: several ways to reduce each term. The remainder is unique if G is a Gr¨ obner basis. The quotients depend on a selection strategy . n / 2 quotients of degree d n / 4 quotients of degree 2 d n / 8 quotients of degree 4 d . . . = ⇒ The degree of the quotients is controlled. Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 8 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Truncated basis elements What is 125 231 546 432 quo 12 358 748 151 ? Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 9 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Truncated basis elements What is 125 231 546 432 quo 12 358 748 151 ? If we know the size of the quotient, then only a few head terms are relevant. Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 9 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Truncated basis elements What is 125 231 546 432 quo 12 358 748 151 ? If we know the size of the quotient, then only a few head terms are relevant. Q 3 G 3 Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 9 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Truncated basis elements What is 125 231 546 432 quo 12 358 748 151 ? If we know the size of the quotient, then only a few head terms are relevant. G # Q 3 3 With the dichotomic selection strategy, G # := ( G # 0 , . . . , G # n ) requires only ˜ O ( n 2 ) coefficients. Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 9 / 27

Key ingredients Dichotomic selection strategy Vanilla Gr¨ obner bases Truncated basis elements Case of the grevlex order Rewriting the equation Rewriting the equation P − � Q i G i ≈ P − � Q i G # up to a certain precision. We need to i increase this precision to continue the computation. Joris van der Hoeven and Robin Larrieu Reduction for generic bivariate ideals 10 / 27