On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
Polynomial Instances of the Positive Semidefinite and Euclidean Distance Matrix Completion Problems
SIAM Journal on Matrix Analysis and Applications
On the Complexity of Semidefinite Programs
Journal of Global Optimization
Proceedings of the 11th Colloquium on Automata, Languages and Programming
Computing integral points in convex semi-algebraic sets
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Computing sum of squares decompositions with rational coefficients
Theoretical Computer Science
Complexity of integer quasiconvex polynomial optimization
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Journal of Symbolic Computation
Deciding reachability of the infimum of a multivariate polynomial
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Computing rational solutions of linear matrix inequalities
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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Let $\mathcal{P}=\{h_1,\dots,h_s\}\subset\mathbb{Z}[Y_1,\dots,Y_k]$, $D\geq\deg(h_i)$ for $1\leq i\leq s$, $\sigma$ bounding the bit length of the coefficients of the $h_i$'s, and let $\Phi$ be a quantifier-free $\mathcal{P}$-formula defining a convex semialgebraic set. We design an algorithm returning a rational point in $\mathcal{S}$ if and only if $\mathcal{S}\cap\mathbb{Q}\neq\emptyset$. It requires $\sigma^{\mathrm{O}(1)}D^{\mathrm{O}(k^3)}$ bit operations. If a rational point is outputted, its coordinates have bit length dominated by $\sigma D^{\mathrm{O}(k^3)}$. Using this result, we obtain a procedure for deciding whether a polynomial $f\in\mathbb{Z}[X_1,\dots,X_n]$ is a sum of squares of polynomials in $\mathbb{Q}[X_1,\dots,X_n]$. 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 $\tau^{\mathrm{O}(1)}D^{\mathrm{O}(k^3)}$ bit operations, and the coefficients of the outputted polynomials have bit length dominated by $\tau D^{\mathrm{O}(k^3)}$.