Skip to Main content Skip to Navigation

Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions

Abstract : Let ${\cal P}=\{h_1, \ldots, h_s\}\subset \Z[Y_1, \ldots, Y_k]$, $D\geq \deg(h_i)$ for $1\leq i \leq s$, $\sigma$ bounding the bit length of the coefficients of the $h_i$'s, and $\Phi$ be a quantifier-free ${\cal P}$-formula defining a convex semi-algebraic set. We design an algorithm returning a rational point in ${\cal S}$ if and only if ${\cal S}\cap \Q\neq\emptyset$. It requires $\sigma^{\bigO(1)}D^{\bigO(k^3)}$ bit operations. If a rational point is outputted its coordinates have bit length dominated by $\sigma D^{\bigO(k^3)}$. Using this result, we obtain a procedure deciding if a polynomial $f\in \Z[X_1, \ldots, X_n]$ is a sum of squares of polynomials in $\Q[X_1, \ldots, X_n]$. Denote by $d$ the degree of $f$, $\tau$ the maximum bit length of the coefficients in $f$, $D={{n+d}\choose{n}}$ and $k\leq D(D+1)-{{n+2d}\choose{n}}$. This procedure requires $\tau^{\bigO(1)}D^{\bigO(k^3)}$ bit operations and the coefficients of the outputted polynomials have bit length dominated by $\tau D^{\bigO(k^3)}$.
Complete list of metadatas

Cited literature [21 references]  Display  Hide  Download

https://hal.inria.fr/inria-00419983
Contributor : Mohab Safey El Din <>
Submitted on : Thursday, October 15, 2009 - 7:22:30 PM
Last modification on : Wednesday, May 15, 2019 - 3:43:57 AM
Document(s) archivé(s) le : Tuesday, June 15, 2010 - 10:16:19 PM

Files

RR-7045.pdf
Files produced by the author(s)

Identifiers

Citation

Mohab Safey El Din, Lihong Zhi. Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions. SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2010, 20 (6), pp.2876-2889. ⟨10.1137/090772459⟩. ⟨inria-00419983⟩

Share

Metrics

Record views

617

Files downloads

423