Section: New Results

Fundamental algorithms and structured polynomial systems

Sparse Gröbner Bases

Sparse elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a semigroup algebra, i.e. an algebra generated by a subset of Laurent monomials. In order to solve symbolically sparse systems, we introduce sparse Gröbner bases, an analog of classical Gröbner bases for semigroup algebras, and we propose sparse variants of the $F_5$ and FGLM algorithms to compute them.

In the case where the generating subset of monomials corresponds to the points with integer coordinates in a normal lattice polytope $𝒫\subset {ℝ}^n$ and under regularity assumptions, we prove in [19] complexity bounds which depend on the combinatorial properties of $𝒫$. These bounds yield new estimates on the complexity of solving 0-dim systems where all polynomials share the same Newton polytope (unmixed case). For instance, we generalize the bound $min(n_1,n_2)+1$ on the maximal degree in a Gröbner basis of a 0-dim. Bilinear system with blocks of variables of sizes $(n_1,n_2)$ to the multihomogeneous case: $n+2-{max}_i\left(\lceil (n_i+1)/d_i\rceil \right)$. We also propose a variant of Fröberg's conjecture which allows us to estimate the complexity of solving overdetermined sparse systems.

Moreover, our prototype “proof-of-concept” implementation shows large speed-ups (more than 100 for some examples) compared to optimized (classical) Gröbner bases software.

Gröbner bases for weighted homogeneous systems

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

Gröbner bases for weighted homogeneous systems can be computed by adapting existing algorithms for homogeneous systems to the weighted homogeneous case. In [29] , we show that in this case, the complexity estimate for Algorithm F5 $\left({\left(\genfrac{}{}{0pt}{}{n+d_{max}-1}{d_{max}}\right)}^{\omega }\right)$ can be divided by a factor ${\left(\prod w_i\right)}^{\omega }$. For zero-dimensional systems, the complexity of Algorithm FGLM $n{D}^{\omega }$ (where $D$ is the number of solutions of the system) can be divided by the same factor ${\left(\prod w_i\right)}^{\omega }$. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of $W$-degree $(d_1,\cdots ,d_n)$, 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 is bounded by the weighted Macaulay bound $\sum (d_i-w_i)+w_n$, 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.

Computing necessary integrability conditions for planar parametrized homogeneous potentials

Let $V\in \mathbb{Q}\left(i\right)({𝐚}_1,\cdots ,{𝐚}_n)({𝐪}_1,{𝐪}_2)$ be a rationally parametrized planar homogeneous potential of homogeneity degree $k\ne -2,0,2$. In [12] , we design an algorithm that computes polynomial necessary conditions on the parameters $({𝐚}_1,\cdots ,{𝐚}_n)$ such that the dynamical system associated to the potential $V$ is integrable. These conditions originate from those of the Morales-Ramis-Simó integrability criterion near all Darboux points and make use of Gröbner bases algorithms. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree 9. Another striking application is the first complete proof of the non-integrability of the collinear three body problem.