Dima Grigoriev

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Two important algebraic proof systems are the Nullstellensatz system [1] and the polynomial calculus [2] (also called the Gröbner system). The Nullstellensatz system is a propositional proof system based on Hilbert’s Nullstellensatz, and the polynomial calculus (PC) is a proof system which allows derivations of polynomials, over some £eld. The complexity of(More)
The problem of factoring a linear partial differential operator is studied. An algorithm is designed which allows one to factor an operator when its symbol is separable, and if in addition the operator has enough right factors then it is completely reducible. Since finding the space of solutions of a completely reducible operator reduces to the same for its(More)
A lower bound is established on degrees of Positivstellensatz calculus refutations (over a real field) introduced in (Grigoriev & Vorobjov 2001; Grigoriev 2001) for the knapsack problem. The bound depends on the values of coefficients of an instance of the knapsack problem: for certain values the lower bound is linear and for certain values the upper bound(More)
We introduce two versions of proof systems dealing with systems of inequalities: Positivstellensatz refutations and Positivstellensatz calculus. For both systems we prove the lower bounds on degrees and lengths of derivations for the example due to Lazard, Mora and Philippon. These bounds are sharp, as well as they are for the Nullstellensatz refutations(More)
Recall that a polynomial f 2 FX1; : : : ; Xn] is t-sparse, if f = P IX I contains at most t terms. In BT 88], GKS 90] (see also GK 87] and Ka 89]) the problem of interpolation of t-sparse polynomial given by a black-box for its evaluation has been solved. In this paper we shall assume that F is a eld of characteristic zero. One can consider a t-sparse(More)