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An optimal bound for quasiconvex integer programming. (Une borne optimale pour la programmation entière quasi-convexe.) (French) Zbl 0777.90034
Summary: Let $$F_ 1,\dots,F_ s\in\mathbb{Z}[X_ 1,\dots,X_ n]$$ be quasiconvex polynomials of degrees bounded by $$d\geq 2$$ and assume that the maximum binary length of the coefficients of these polynomials doesn’t exceed a given natural number $$\ell$$. We show that the system of polynomial inequalities $$F_ 1\leq 0,\dots,F_ s\leq 0$$ admits an integer solution if and only if such a solution with binary length bounded by $$(sd)^{cn}\ell$$ exists. (Here $$c$$ is a constant, independent on $$s$$, $$d$$, $$n$$ and $$\ell$$). The simply exponential nature of this bound is intrinsical to this problem. We obtain a similar geometrical bound for the corresponding minimization problem.
##### MSC:
 90C10 Integer programming 65K05 Numerical mathematical programming methods
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##### References:
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