Almost linear time operations with triangular sets

Abstract

Let F be a perfect field, and let X = X1, . . . , Xn be indeterminates over F. A (monic) triangular set T = (T1, . . . , Tn) is a family of polynomials in F[X] such that for all i, Ti is in F[X1, . . . , Xi], monic inXi, and reduced modulo 〈T1, . . . , Ti−1〉. The degree of T is the product deg(T1, X1) · · · deg(Tn, Xn). These objects allow one to solve a variety of problems for systems of polynomial equations, see [7, 1, 10, 6, 12]. We are interested here in the complexity of operations modulo a given triangular set T. The first question is modular multiplication: given polynomials A,B reduced modulo T, compute AB mod T. Further operations involve families of triangular sets. The lexicographic Grobner basis of an ideal I for a given variable order may not be triangular. The workaround is to decompose I as I = I1∩· · ·∩Is, with pairwise coprime Ij, where each Ij admits a triangular basis. The decomposition is in general not unique, but there exists a canonical choice, the equiprojectable decomposition [4]. That said, the most useful notion of “inversion” is quasi-inverses: given A reduced modulo T, we decompose the ideal 〈T〉 as I0 ∩ I1, where A is zero modulo I0 and invertible modulo I1; the output is the equiprojectable decompositions of I0, I1, and the inverse of A modulo the triangular sets that define I1. The next question is change of order: starting from T, we output the equiprojectable decomposition of the ideal 〈T〉, for a new order on the variables. The last question starts from a family T, . . . ,T which generate pairwise coprime ideals; our output is the equiprojectable decomposition of the ideal they generate. The following theorem provides quasi-linear time results for these questions. These results are valid over a finite field, with costs given in a boolean RAM model; the algorithms are Las Vegas. The main idea is to introduce a primitive element and change representation, as most problems above can be solved easily in univariate situations. The change of representation is done using algorithms for modular composition [3] and power projection [13], but in multivariate setting. In [8], Kedlaya and Umans introduced quasi-linear time algorithms for the univariate versions of these problems; our core technical ingredients are multivariate versions of their algorithms.