  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.
