Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gr¨ obner Bases Joris van der Hoeven, Robin Larrieu Laboratoire d’Informatique de l’Ecole Polytechnique (LIX) ISSAC ’18 – New York, USA 18 / 07 / 2018 Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
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 Fast reduction of bivariate polynomials
Introduction Fast Gr¨ obner basis algorithms rely on linear algebra (ex: F4, F5. . . ) → Not optimal unless ω = 2. Can we do it with polynomial arithmetic? → Hope for asymptotically optimal algorithms. Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Introduction Fast Gr¨ obner basis algorithms rely on linear algebra (ex: F4, F5. . . ) → Not optimal unless ω = 2. Can we do it with polynomial arithmetic? → Hope for asymptotically optimal algorithms. Easier problem Given a Gr¨ obner basis G , can we reduce P modulo G faster? Main result If G is sufficiently regular, a quasi-optimal algorithm exists modulo precomputation. Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Polynomial reduction: complexity Y X I := � 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 Fast reduction of bivariate polynomials
Outline Vanilla Gr¨ obner bases 1 Definition Terse representation Polynomial reduction 2 Idea of the algorithm Applications Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Outline Vanilla Gr¨ obner bases 1 Definition Terse representation Polynomial reduction 2 Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Definition We consider the term orders ≺ k ( k ∈ N ∗ ) as the weighted-degree lexicographic order with weights ( X : 1 , Y : k ). Vanilla Gr¨ obner stairs The monomials below the stairs are the minimal elements with respect to ≺ k Example for k = 4 and an ideal I of degree D = 237 Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Definition (2) Retractive property let I := { 0 , 1 , n } . The retractive property means that for any i � n we have a linear combination � G i = C i , j G j . j ∈ I Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Definition (2) Retractive property For ℓ ∈ N ∗ , let I ℓ := { 0 , 1 , n } ∪ ℓ N ∩ (0 , n ). The retractive property means that for any i , ℓ � n we have a linear combination � G i = C i , j ,ℓ G j with deg k C i , j ,ℓ = O ( kl ) . j ∈ I ℓ Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Definition (2) Retractive property For ℓ ∈ N ∗ , let I ℓ := { 0 , 1 , n } ∪ ℓ N ∩ (0 , n ). The retractive property means that for any i , ℓ � n we have a linear combination � G i = C i , j ,ℓ G j with deg k C i , j ,ℓ = O ( kl ) . j ∈ I ℓ More precisely, deg k C i , j ,ℓ < k (2 ℓ − 1) . Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Definition (2) Retractive property For ℓ ∈ N ∗ , let I ℓ := { 0 , 1 , n } ∪ ℓ N ∩ (0 , n ). The retractive property means that for any i , ℓ � n we have a linear combination � G i = C i , j ,ℓ G j with deg k C i , j ,ℓ = O ( kl ) . j ∈ I ℓ More precisely, deg k C i , j ,ℓ < k (2 ℓ − 1) . A Gr¨ obner basis for the k -order is vanilla if it is a vanilla Gr¨ obner stairs and has the retractive property. Conjecture: vanilla Gr¨ obner bases are generic Experimentally, for generators chosen at random, and for various term orders, the Gr¨ obner basis is vanilla. Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Terse representation G 0 , G 1 , G n and well-chosen retraction coefficients hold all information (in space ˜ O ( n 2 )) and allow to retrieve G fast. The coefficients of each G i are needed to compute the reduction, but there are too many. Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Terse representation G 0 , G 1 , G n and well-chosen retraction coefficients hold all information (in space ˜ O ( n 2 )) and allow to retrieve G fast. The coefficients of each G i are needed to compute the reduction, but there are too many. = ⇒ Keep only enough coefficients to evaluate Q i . Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Terse representation G 0 , G 1 , G n and well-chosen retraction coefficients hold all information (in space ˜ O ( n 2 )) and allow to retrieve G fast. The coefficients of each G i are needed to compute the reduction, but there are too many. = ⇒ Keep only enough coefficients to evaluate Q i . = ⇒ Control the degree of the quotients. Dichotomic selection strategy n / 2 quotients of degree d . n / 4 quotients of degree 2 d . n / 8 quotients of degree 4 d . . . . Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Terse representation – Example G 0 G 1 G 2 G 3 + the linear combination + the linear combination G 2 = f 2 ( G i , i ∈ { 0 , 1 , 4 , 8 , 11 } ) G 3 = f 3 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) (5 polynomials of degree 27) (8 polynomials of degree 11) G 4 G 5 G 6 G 7 + the linear combination + the linear combination + the linear combination + the linear combination G 4 = f 4 ( G i , i ∈ { 0 , 1 , 8 , 11 } ) G 5 = f 5 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) G 6 = f 6 ( G i , i ∈ { 0 , 1 , 4 , 8 , 11 } ) G 7 = f 7 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) (4 polynomials of degree 59) (8 polynomials of degree 11) (5 polynomials of degree 27) (8 polynomials of degree 11) G 8 G 9 G 10 G 11 + the linear combination + the linear combination + the linear combination G 8 = f 8 ( G i , i ∈ { 0 , 1 , 11 } ) G 9 = f 9 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) G 10 = f 10 ( G i , i ∈ { 0 , 1 , 4 , 8 , 11 } ) (3 polynomials of degree 123) (8 polynomials of degree 11) (5 polynomials of degree 27) Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Definition Polynomial reduction Terse representation Terse representation – Example G # G # G # G # 0 1 2 3 + the linear combination + the linear combination G 2 = f 2 ( G i , i ∈ { 0 , 1 , 4 , 8 , 11 } ) G 3 = f 3 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) (5 polynomials of degree 27) (8 polynomials of degree 11) G # G # G # G # 4 5 6 7 + the linear combination + the linear combination + the linear combination + the linear combination G 4 = f 4 ( G i , i ∈ { 0 , 1 , 8 , 11 } ) G 5 = f 5 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) G 6 = f 6 ( G i , i ∈ { 0 , 1 , 4 , 8 , 11 } ) G 7 = f 7 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) (4 polynomials of degree 59) (8 polynomials of degree 11) (5 polynomials of degree 27) (8 polynomials of degree 11) G # G # G # G # 8 9 10 11 + the linear combination + the linear combination + the linear combination G 8 = f 8 ( G i , i ∈ { 0 , 1 , 11 } ) G 9 = f 9 ( G i , i ∈ { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 11 } ) G 10 = f 10 ( G i , i ∈ { 0 , 1 , 4 , 8 , 11 } ) (3 polynomials of degree 123) (8 polynomials of degree 11) (5 polynomials of degree 27) Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Idea of the algorithm Polynomial reduction Applications Outline Vanilla Gr¨ obner bases 1 Polynomial reduction 2 Idea of the algorithm Applications Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Vanilla Gr¨ obner bases Idea of the algorithm Polynomial reduction Applications Idea of the algorithm Theorem (van der Hoeven – ACA 2015) Using relaxed multiplications, 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 O ( n 3 ) and we would like to achieve ˜ O ( n 2 ). Joris van der Hoeven and Robin Larrieu Fast reduction of bivariate polynomials
Recommend
More recommend
