On the construction of staggered linear bases

A staggered linear basis is a specific basis for a polynomial ideal generating the ideal as a linear vector space. There are some algorithms in the literature to compute these bases and we show that they do not work properly. In this paper, due to the strong connection between Gröbner bases and staggered linear bases and by applying a signature-based structure, we present a new and correct algorithm to compute staggered linear bases. Since the output of this algorithm also remains a Gröbner basis for its input ideal, we conclude the paper by discussing the performance of this algorithm compared with the newest signature-based algorithm to compute Gröbner bases.

The hypergeometric system for one-loop triangle integral

We derive the holonomic hypergeometric system for the [Formula: see text] function with two equal virtual masses, and present the expression of [Formula: see text] in hypergeometric series in corresponding convergent region. Combining the Horn’s convergence theory with Gröbner basis of polynomial ideal, one can calculate the convergence region of the corresponding multiple series concretely. Using the system given here, one can analytically continue [Formula: see text] to whole parameter space.