Paper detail

Faster change of order algorithm for Gröbner bases under shape and stability assumptions

Solving zero-dimensional polynomial systems using Gröbner bases is usually done by, first, computing a Gröbner basis for the degree reverse lexicographic order, and next computing the lexicographic Gröbner basis with a change of order algorithm. Currently, the change of order now takes a significant part of the whole solving time for many generic instances. Like the fastest known change of order algorithms, this work focuses on the situation where the ideal defined by the system satisfies natural properties which can be recovered in generic coordinates. First, the ideal has a \emph{shape} lexicographic Gröbner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a \emph{stability} property; in particular, the multiplication matrix can be read on the input Gröbner basis. The current fastest algorithms rely on the sparsity of this matrix. Actually, this sparsity is a consequence of an algebraic structure, which can be exploited to represent the matrix concisely as a univariate polynomial matrix. We show that the Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case. Under some mild assumption implying $n \le t$, the arithmetic complexity of our algorithm is $O\tilde{~}(t^{ω-1}D)$, where $n$ is the number of variables, $t$ is a sparsity indicator of the aforementioned matrix, $D$ is the degree of the zero-dimensional ideal under consideration, and $ω$ is the exponent of matrix multiplication. This improves upon both state-of-the-art complexity bounds $O\tilde{~}(tD^2)$ and $O\tilde{~}(D^ω)$, since $ω< 3$ and $t\le D$. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit.