On the complexity of computing Gröbner bases for weighted homogeneous systems. (September 2016)

• Record Type: Journal Article
• Authors: Faugère, Jean-Charles; Safey El Din, Mohab; Verron, Thibaut
• Is Part Of: Journal of symbolic computation. Volume 76(2016)
• Page Start: 107
• Page End: 141
• Publication Date: 2016-09
• Subjects: Gröbner bases -- Polynomial system solving -- Quasi-homogeneous systems -- Weighted homogeneous systems; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.12.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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 deg W ⁡ ( X 1 α 1 … X n α n ) = ∑ w i α i . 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 F 5 ( ( n + d max − 1 d max ) ω ) can be divided by a factor ( ∏ w i ) ω . For zero-dimensional systems, the complexity of AlgorithmFGLM n D 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 n d i / ∏ i = 1 n w i . Furthermore, the maximum degree reached in a run of Algorithm F 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 can yield substantial speed-ups, and allows us to solve systems which were otherwise out of reach.