Almost linear time operations with triangular sets

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 in Xi, 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 Gröbner 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 equiprojectabl