Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals

Abstract

Let A, B ∈ [X, Y] be two bivariate polynomials over an effective field , and let G be the reduced Gröbner basis of the ideal I ≔ ⟨A, B⟩ generated by A and B with respect to the usual degree lexicographic order. Assuming A and B sufficiently generic, we design a quasi-optimal algorithm for the reduction of P ∈ [X, Y] modulo G, where "quasi-optimal" is meant in terms of the size of the input A, B, P. Immediate applications are an ideal membership test and a multiplication algorithm for the quotient algebra ≔ [X, Y]/⟨A, B⟩, both in quasi-linear time. Moreover, we show that G itself can be computed in quasi-linear time with respect to the output size. Operation Our paper Previous best Deglex Gröbner basis G O(R(m 2 ) + R(n m) n log n) = Õ (|A, B, G|) O(n 2 R(n m)) Structure of [X, Y]/⟨A, B⟩ O(R(m 2 ) + M(n m) log n) = Õ (|A, B|) O(n 2 R(n m)) Reduction of P with deg P = d O(R(d 2 ) + R(n m) log n) = Õ (|P| + D) O(R(d 2 ) + n R(n m)) Multiplication in [X, Y]/⟨A, B⟩ O(R(n m) log n) = Õ (D)

High-order lifting for polynomial Sylvester matrices

“…The algorithm in [16] resulted in the first quasi-linear complexity bound over finite fields F for sufficiently generic p and q, with respect to the total degree. It relies on the concise representation of a Gröbner basis of the bivariate ideal I = ⟨p, q⟩ [15]. Such a representation has size Õ(n 2 ) elements in F, and allows multiplication in K[x, y]/I in quasi-linear time [15].…”

“…It relies on the concise representation of a Gröbner basis of the bivariate ideal I = ⟨p, q⟩ [15]. Such a representation has size Õ(n 2 ) elements in F, and allows multiplication in K[x, y]/I in quasi-linear time [15]. This fast multiplication then leads to an efficient reduction of the resultant problem to a bivariate modular composition problem, in turn reduced to a multivariate multipoint evaluation problem [16].…”

mwp-Analysis Improvement and Implementation: Realizing Implicit Computational Complexity

“…if α β and both α × γ and β × γ are non-zero, then α × γ β × γ. This order is leveraged to obtain efficient algorithms, similar to what is done using Gröbner bases for computation of standard polynomials [31]. For instance, the algorithm for multiplication of polynomials uses this property to compute the product of an ordered polynomial P with n i=1 α i ki j=1 δ(a i,j , b i,j ):…”

Faster Modular Composition

“…For an ideal given by two generic bivariate polynomials of degree 𝑛 (hence the ideal is of degree 𝑛 2 ) and the graded lexicographic order, van der Hoeven and Larrieu avoid the use of an explicit Gröbner basis. They show that a concise representation of the basis of size only Õ (𝑛 2 ) is sufficient for reducing a polynomial modulo the ideal in time Õ (𝑛 2 ) [29]; the concise representation consists in particular of truncations of well chosen polynomials in the ideal. It is unclear to us whether a similar truncation strategy could be applied specifically to I, whose degree is only 𝑛.…”