On the complexity of computing Gröbner bases for weighted homogeneous systems

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• 作者：Jean-Charles Faugè ; re^c ; ^a ; ^b ; ^{Jean-Charles.Faugere@inria.fr" class="auth_mail" title="E-mail the corresponding author} ; Mohab Safey El Din^a ; ^b ; ^c ; ^d ; ^{Mohab.Safey@lip6.fr" class="auth_mail" title="E-mail the corresponding author} ; Thibaut Verron^a ; ^b ; ^c ; ^{Thibaut.Verron@lip6.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Grö ; bner bases ; Polynomial system solving ; Quasi-homogeneous systems ; Weighted homogeneous systems
• 刊名：Journal of Symbolic Computation
• 出版年：2016
• 出版时间：September-October 2016
• 年：2016
• 卷：76
• 期：Complete
• 页码：107-141
• 全文大小：719 K

文摘

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,…,w_n)$W=(w_1,\dots ,w_n)$, W  -homogeneous polynomials are polynomials which are homogeneous w.r.t. the weighted degree ${\mathrm{deg}}_W(X_1^{{\alpha }_1}\dots X_n^{{\alpha }_n})=\sum w_i{\alpha }_i$.

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 F₅${\mathsf{F}}_{\mathsf{5}}$$\left({\left(\begin{array}{c}\hfill n+d_{\max }-1\hfill \\ \hfill d_{\max }\hfill \end{array}\right)}^{\omega }\right)$ can be divided by a factor (∏w_i)^ω${(\prod w_i)}^{\omega }$. For zero-dimensional systems, the complexity of Algorithm FGLMnD^ω$n{D}^{\omega }$ (where D   is the number of solutions of the system) can be divided by the same factor (∏w_i)^ω${(\prod w_i)}^{\omega }$. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of W  -degree (d₁,…,d_n)$(d_1,\dots ,d_n)$, these complexity estimates are polynomial in the weighted Bézout bound ${\prod }_{i=1}^n{d}_i/{\prod }_{i=1}^n{w}_i$.

Furthermore, the maximum degree reached in a run of Algorithm F₅${\mathsf{F}}_{\mathsf{5}}$ is bounded by the weighted Macaulay bound ∑(d_i−w_i)+w_n$\sum (d_i-w_i)+w_n$, and this bound is sharp if we can order the weights so that w_n=1$w_n=1$. 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.