Real root finding for equivariant semi-algebraic systems

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most $2d-1$ distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$ in time $(sn)^{O(d)}$. This improves the state-of-the-art which is exponential in $n$. When the variables $x_1, \ldots, x_n$ are quantified and the coefficients of the input system depend on parameters $y_1, \ldots, y_t$, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time $(sn)^{O(dt)}$

Minimizing Rational Functions by Exact Jacobian SDP Relaxation Applicable to Finite Singularities

This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as polynomial optimization by the technique of homogenization. These two problems are shown to be equivalent under some generic conditions. The exact Jacobian SDP relaxation method proposed by Nie is used to solve the resulting polynomial optimization. We also prove that the assumption of nonsingularity in Nie's method can be weakened as the finiteness of singularities. Some numerical examples are given to illustrate the efficiency of our method.Comment: 23 page

On the Complexity of the Generalized MinRank Problem

We study the complexity of solving the \emph{generalized MinRank problem}, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a \emph{determinantal ideal}: the ideal generated by all minors of size $r+1$ of the matrix. We give new complexity bounds for solving this problem using Gr\"obner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0-dimensional and radical system of bi-degree $(D,1)$. We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.Comment: 29 page

Real root finding for equivariant semi-algebraic systems

International audienceLet $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most $2d-1$ distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$ in time $(sn)^{O(d)}$. This improves the state-of-the-art which is exponential in $n$. When the variables $x_1, \ldots, x_n$ are quantified and the coefficients of the input system depend on parameters $y_1, \ldots, y_t$, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time $(sn)^{O(dt)}$

Global optimization of polynomials restricted to a smooth variety using sums of squares

International audienceLet $f_1,\dots,f_p$ be in \Q[\bfX], where \bfX=(X_1,\dots,X_n)^t, that generate a radical ideal and let $V$ be their complex zero-set. Assume that $V$ is smooth and equidimensional. Given f\in\Q[\bfX] bounded below, consider the optimization problem of computing $f^\star=\inf_{x\in V\cap\R^n} f(x)$. For \bfA\in GL_n(\C), we denote by f^\bfA the polynomial f(\bfA\bfX) and by V^\bfA the complex zero-set of f_1^\bfA,\ldots,f_p^\bfA. We construct families of polynomials {\sf M}^\bfA_0, \ldots, {\sf M}^\bfA_d in \Q[\bfX]: each {\sf M}_i^\bfA is related to the section of a linear subspace with the critical locus of a linear projection. We prove that there exists a non-empty Zariski-open set \mathscr{O}\subset GL_n(\C) such that for all \bfA\in \mathscr{O}\cap GL_n(\Q), $f(x)$ is non-negative for all $x\in V\cap\R^n$ if, and only if, f^\bfA can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal \langle {\sf M}^\bfA_i\rangle, for $0\leq i \leq d$. Hence, we can obtain algebraic certificates for lower bounds on $f^\star$ using semidefinite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in {\sf M}^\bfA_i