On the complexity of computing Gr{ö}bner bases for weighted homogeneous systems

نویسندگان

  • Jean-Charles Faugère
  • Mohab Safey El Din
  • Thibaut Verron
چکیده

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 = (w1, . . . , wn), W -homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree degW (X α1 1 . . . X αn n ) = ∑ wiαi. Gröbner bases for weighted homogeneous systems can be computed by adapting existing algorithms for homogeneous systems to the weighted homogeneous case. We show that in this case, the complexity estimate for Algorithm F5 ( ( n+dmax−1 dmax )ω ) can be divided by a factor ( ∏ wi) . For zero-dimensional systems, the complexity of Algorithm FGLM nD (where D is the number of solutions of the system) can be divided by the same factor ( ∏ wi) . Under genericity assumptions, for zero-dimensional weighted homogeneous systems of W -degree (d1, . . . , dn), these complexity estimates are polynomial in the weighted Bézout bound ∏n i=1 di/ ∏n i=1 wi. Furthermore, the maximum degree reached in a run of Algorithm F5 is bounded by the weighted Macaulay bound ∑ (di−wi)+wn, and this bound is sharp if we can order the weights so that wn = 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 can yield substantial speed-ups, and allows us to solve systems which were otherwise out of reach. Email addresses: [email protected] (Jean-Charles Faugère), [email protected] (Mohab Safey El Din), [email protected] (Thibaut Verron). Preprint submitted to Journal of Symbolic Computation December 22, 2015

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عنوان ژورنال:
  • J. Symb. Comput.

دوره 76  شماره 

صفحات  -

تاریخ انتشار 2016