On the Complexity of Computing Gröbner Bases for Quasi-homogeneous Systems

Abstract

Let $\K$ be a field and $(f_1, \ldots, f_n)\subset \K[X_1, \ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1,\dots ,d_n)$ w.r.t a system of weights $(w_1,\dots ,w_n)$. Such systems are likely to arise from a lot of applications, including physics or cryptography. We design strategies for computing Gröbner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi-homogeneous system, the complexity of the full strategy is polynomial in the weighted Bézout bound ${\prod }_{i=1}^n{d}_i/{\prod }_{i=1}^n{w}_i$. We provide some experimental results based on generic systems as well as systems arising from a cryptography problem. They show that taking advantage of the quasi-homogeneous structure of the systems allow us to solve systems that were out of reach otherwise.