Fast Gr obner basis computation and polynomial reduction for - PowerPoint PPT Presentation

Fast Gr¨ obner basis computation and polynomial reduction for generic bivariate ideals Joris van der Hoeven, Robin Larrieu Laboratoire d’Informatique de l’Ecole Polytechnique (LIX) JNCF 2019 – Luminy 05th February 2019 Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 1 / 17

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 Generic bivariate ideals 2 / 17

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 Generic bivariate ideals 2 / 17

Introduction 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 Generic bivariate ideals 3 / 17

Introduction 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 Generic bivariate ideals 3 / 17

Introduction 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 Generic bivariate ideals 3 / 17

Introduction 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? Setting and notations I = � A , B � with generic A , B ∈ K [ X , Y ] given in total degree. Use the degree lexicographic order to compute G . deg A = n and deg B = m with n � m (in this talk n = m ) We want to reduce P with deg P = d Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 3 / 17

Introduction 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? Setting and notations I = � A , B � with generic A , B ∈ K [ X , Y ] given in total degree. Use the degree lexicographic order to compute G . deg A = n and deg B = m with n � m (in this talk n = m ) We want to reduce P with deg P = d Main result In this specific setting, a quasi-optimal algorithm exists ! Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 3 / 17

Outline Presentation of the problem 1 Polynomial reduction: complexity Gr¨ obner bases: concise representation Faster computation 2 Polynomial reduction Gr¨ obner basis Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 4 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Outline Presentation of the problem 1 Polynomial reduction: complexity Gr¨ obner bases: concise representation Faster computation 2 Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 5 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Polynomial reduction: complexity Y X A , B : Θ( n 2 ) coefficients K [ X , Y ] / I : dimension Θ( n 2 ) G : Θ( n 3 ) coefficients (Θ( n 2 ) for each G i ) Reduction using G needs at least Θ( n 3 ) = ⇒ reduction with less information? Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 6 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Related results Theorem (van der Hoeven – ACA 2015) The extended reduction of P modulo G can be computed in quasi-linear time with respect to the size of the equation � P = Q i G i + R i Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 7 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Related results Theorem (van der Hoeven – ACA 2015) The extended reduction of P modulo G can be computed in quasi-linear time with respect to 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 Generic bivariate ideals 7 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Related results Theorem (van der Hoeven – ACA 2015) The extended reduction of P modulo G can be computed in quasi-linear time with respect to 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 Generic bivariate ideals 7 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Related results Theorem (van der Hoeven, L. – ISSAC 2018) A special class of bases called vanilla Gr¨ obner bases admit a terse representation in ˜ O ( n 2 ) space. Assuming this representation has been precomputed, reduction can be done in time ˜ O ( n 2 ). Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 8 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Related results Theorem (van der Hoeven, L. – ISSAC 2018) A special class of bases called vanilla Gr¨ obner bases admit a terse representation in ˜ O ( n 2 ) space. Assuming this representation has been precomputed, reduction can be done in time ˜ O ( n 2 ). Problem: in this setting, G is not vanilla. (vanilla Gr¨ obner bases rely on different assumptions) Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 8 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Related results Theorem (van der Hoeven, L. – ISSAC 2018) A special class of bases called vanilla Gr¨ obner bases admit a terse representation in ˜ O ( n 2 ) space. Assuming this representation has been precomputed, reduction can be done in time ˜ O ( n 2 ). Problem: in this setting, G is not vanilla. (vanilla Gr¨ obner bases rely on different assumptions) But . . . similar ideas still apply. (We use essentially the same tricks, although the algorithm is very different). Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 8 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Gr¨ obner bases: concise representation – 1 The Gr¨ obner basis is generated by A and B = ⇒ there are relations between the G i (redundant information) Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 9 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Gr¨ obner bases: concise representation – 1 The Gr¨ obner basis is generated by A and B = ⇒ there are relations between the G i (redundant information) Reduced Gr¨ obner basis: G red i +2 = Spol ( G red , G red i +1 ) rem G red 0 , . . . , G red i i +1 Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 9 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Gr¨ obner bases: concise representation – 1 The Gr¨ obner basis is generated by A and B = ⇒ there are relations between the G i (redundant information) Reduced Gr¨ obner basis: G red i +2 = Spol ( G red , G red i +1 ) rem G red 0 , . . . , G red i i +1 Remark: G i +2 = Spol ( G i , G i +1 ) rem G i , G i +1 also gives a Gr¨ obner basis. Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 9 / 17

Presentation of the problem Polynomial reduction: complexity Faster computation Gr¨ obner bases: concise representation Gr¨ obner bases: concise representation – 1 The Gr¨ obner basis is generated by A and B = ⇒ there are relations between the G i (redundant information) Reduced Gr¨ obner basis: G red i +2 = Spol ( G red , G red i +1 ) rem G red 0 , . . . , G red i i +1 Remark: G i +2 = Spol ( G i , G i +1 ) rem G i , G i +1 also gives a Gr¨ obner basis. � G i +1 � � � G i = M i G i +2 G i +1 Joris van der Hoeven and Robin Larrieu Generic bivariate ideals 9 / 17