Sparse Polynomial Arithmetic with the BPAS Library

Abstract

We discuss algorithms for pseudo-division and division with remainder of multivariate polynomials with sparse representation. This work is motivated by the computations of normal forms and pseudo-remainders with respect to regular chains. We report on the implementation of those algorithms with the BPAS library.

A Appendix

Let $\mathbb {K}$ be a field. If B is a Gröbner basis of $\mathbb {K}[x_1, \ldots , x_m]$ Algorithm 5 computes the normal form of a polynomial $a \in \mathbb {K}[x_1, \ldots , x_m]$ (together with the quotients) w.r.t. B; the principle is direct (or naïve). Alternatively, when B is a zero-dimensional normalized (thus so-called Lazard) triangular set, one can use Algorithm 7, the recursive principle of which is taken from [14]. Some details are given here. For computing the normal form of polynomial $a \in \mathbb {K}[x_1,\ldots ,x_m]$ with respect to a Lazard triangular set $T = \{ t_1, \ldots , t_m \} \subset \mathbb {K}[x_1,\ldots ,x_m]$, Algorithm 7 uses the recursive representation of polynomials. If $m = 1$, the result is obtained by applying Algorithm 2. Otherwise, the coefficients of a with respect to $x_m$ are reduced w.r.t. to $\{ t_1, \ldots , t_{m-1} \}$ by means of a recursive call (Lines 4–11 of the pseudo-code), yielding a polynomial r. Then, r is divided by $t_m$ by applying Algorithm 2, see Line 12, yielding a new polynomial r. Finally, the coefficients w.r.t. $x_m$ of this new polynomial r are reduced w.r.t. to $\{ t_1, \ldots , t_{m-1} \}$, by means of a second recursive call, see Lines 13–16.

Algorithm 6 is a direct (or naïve) procedure for computing the pseudo-remainder and the pseudo-quotients of a polynomial $a \in \mathbb {K}[x_1,\ldots ,x_m]$ by a triangular set $T = \{ t_1, \ldots , t_N \}$. Note that T may not be zero-dimensional, that is, $N < m$ may hold. Moreover, T may not be normalized; in particular its initials may not be constant. Algorithm 8 is a recursive version of Algorithm 6 following the same principles as Algorithm 7 and calling Algorithm 4 at Line 14.