On the Termination of the General XL Algorithm and Ordinary Multinomials

The XL algorithm is an algorithm for solving overdetermined systems of multivariate polynomial equations, which was initially introduced for quadratic equations. However, the algorithm works for polynomials of any degree, and in this paper we will focus on the performance of XL for polynomials of degree $\geq3$, where the optimal termination value of the parameter $D$ is still unknown. We prove that the XL algorithm terminates at a certain value of $D$ in the case that the number of equations exceeds the number of variables by 1 or 2. We also give strong evidence that this value is best possible, and we show that this value is smaller than the degree of regularity. Part of our analysis requires proving that ordinary multinomials are strongly unimodal, and this result may be of independent interest.