Section: New Results

Solving systems over the reals and applications

It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a weighted sum of finitely many squares instead of a sum of two squares, then one can choose all coefficients in the representation to lie in the field generated by the coefficients of the polynomial. In particular, this allows an effective treatment of polynomials with rational coefficients. In [11], we describe, analyze and compare both from the theoretical and practical points of view, two algorithms computing such a weighted sums of squares decomposition for univariate polynomials with rational coefficients. The first algorithm, due to the third author relies on real root isolation, quadratic approximations of positive polynomials and square-free decomposition but its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm. They are exponential in the degree of the input univariate polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using quantifier elimination and root isolation bounds. The second algorithm, due to Chevillard, Harrison, Joldes and Lauter, relies on complex root isolation and square-free decomposition and has been introduced for certifying positiveness of poly-nomials in the context of computer arithmetics. Again, its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm, which are polynomial in the degree of the input polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using Vieta's formula and root isolation bounds. Finally, we report on our implementations of both algorithms and compare them in practice on several application benchmarks. While the second algorithm is, as expected from the complexity result, more efficient on most of examples, we exhibit families of non-negative polynomials for which the first algorithm is better.

[9] describes our freely distributed Maple library spectra , for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.

Let $S\subset {ℝ}^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. In [19], we design an algorithm which takes as input a polynomial system defining S and an integer $p\ge 0$ and returns the n-dimensional volume of S at absolute precision $2^{-p}$. Our algorithm relies on the relationship between volumes of semi-algebraic sets and periods of rational integrals. It makes use of algorithms computing the Picard-Fuchs differential equation of appropriate periods, properties of critical points, and high-precision numerical integration of differential equations. The algorithm runs in essentially linear time with respect to $p$. This improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods. Assuming a conjecture of Dimca, the arithmetic cost of the algebraic subroutines for computing Picard-Fuchs equations and critical points is singly exponential in $n$ and polynomial in the maximum degree of the input.

Let $𝐟=(f_1,...,f_s)$ be a sequence of polynomials in $\mathbb{Q}[X_1,...,X_n]$ of maximal degree $D$ and $V\subset {ℂ}^n$ be the algebraic set defined by $𝐟$ and $r$ be its dimension. The real radical $\sqrt{\langle 𝐟\rangle }$ associated to $𝐟$ is the largest ideal which defines the real trace of $V$. When $V$ is smooth, we show in [13], that $\sqrt[re]{\langle 𝐟\rangle }$, has a finite set of generators with degrees bounded by $degV$. Moreover, we present a probabilistic algorithm of complexity ${\left(sn{D}^n\right)}^{O\left(1\right)}$ to compute the minimal primes of $\sqrt[re]{\langle 𝐟\rangle }$. When $V$ is not smooth, we give a probabilistic algorithm of complexity $s^{O\left(1\right)}{\left(nD\right)}^{O\left(nr{2}^r\right)}$ to compute rational parametrizations for all irreducible components of the real algebraic set $V\cap {ℝ}^n$.

Let $(g_1,...,g_p)$ in $\mathbb{Q}[X_1,...,X_n]$ and $S$ be the basic closed semi-algebraic set defined by $g_1\ge 0,...,g_p\ge 0$. The $S$-radical of $\langle 𝐟\rangle$, which is denoted by $\sqrt[S]{\langle 𝐟\rangle }$, is the ideal associated to the Zariski closure of $V\cap S$. We give a probabilistic algorithm to compute rational parametrizations of all irreducible components of that Zariski closure, hence encoding $\sqrt[S]{\langle 𝐟\rangle }$. Assuming now that $D$ is the maximum of the degrees of the $f_i$'s and the $g_i$'s, this algorithm runs in time $2^p{(s+p)}^{O\left(1\right)}{\left(nD\right)}^{O\left(rn{2}^r\right)}$.

Experiments are performed to illustrate and show the efficiency of our approaches on computing real radicals.

In [14], we consider the second-order discontinuous differential equation $y^{\text{'}\text{'}}+\eta \mathrm{sgn}\left(y\right)=\theta y+\alpha sin\left(\beta t\right)$ where the parameters $\eta ,\theta ,\alpha ,\beta$ are real. The main goal is to discuss the existence of periodic solutions. Under explicit conditions, the number of such solutions is given. Furthermore, for each of these periodic solutions, an explicit formula is provided.