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The XL algorithm is an algorithm for solving overdetermined systems ofmultivariate polynomial equations, which was initially introduced for quadraticequations. However, the algorithm works for polynomials of any degree, and inthis paper we will focus on the performance of XL for polynomials of degree$\geq3$, where the optimal termination value of the parameter $D$ is stillunknown. We prove that the XL algorithm terminates at a certain value of $D$ inthe case that the number of equations exceeds the number of variables by 1 or2. We also give strong evidence that this value is best possible, and we showthat this value is smaller than the degree of regularity. Part of our analysisrequires proving that ordinary multinomials are strongly unimodal, and thisresult may be of independent interest.

On the Termination of the General XL Algorithm and Ordinary Multinomials