On the complexity of computing Gröbner bases for weighted homogeneous systems

Jean-Charles Faugère, Mohab Safey El Din, Thibaut Verron
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights W=(w_1,...,w_n), W-homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree _W(X_1^α_1,...,X_n^α_n) = ∑ w_iα_i. Gröbner bases for weighted homogeneous systems can be computed by adapting existing algorithms for homogeneous systems to the weighted
more &raquo; ... geneous case. We show that in this case, the complexity estimate for Algorithm 5 (n+-1^ω) can be divided by a factor (∏ w_i)^ω. For zero-dimensional systems, the complexity of Algorithm nD^ω (where D is the number of solutions of the system) can be divided by the same factor (∏ w_i)^ω. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of W-degree (d_1,...,d_n), these complexity estimates are polynomial in the weighted Bézout bound ∏_i=1^nd_i / ∏_i=1^nw_i. Furthermore, the maximum degree reached in a run of Algorithm 5 is bounded by the weighted Macaulay bound ∑ (d_i-w_i) + w_n, and this bound is sharp if we can order the weights so that w_n=1. For overdetermined semi-regular systems, estimates from the homogeneous case can be adapted to the weighted case. We provide some experimental results based on systems arising from a cryptography problem and from polynomial inversion problems. They show that taking advantage of the weighted homogeneous structure yields substantial speed-ups, and allows us to solve systems which were otherwise out of reach.
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