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Journal Articles SIAM Journal on Optimization Year : 2010

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

Mohab Safey El Din
Lihong Zhi
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#### 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 $\tau D^{\bigO(k^3)}$.

### Dates and versions

inria-00419983 , version 1 (15-10-2009)

### Identifiers

• HAL Id : inria-00419983 , version 1
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### Cite

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

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