We give an algebraic quantifier elimination algorithm for the first-order theory over any given finite field using Gröbner basis meth-ods. The algorithm relies on the strong Nullstellensatz and properties of elimination ideals over finite fields. We analyze the theoretical complex-ity

admits elimination of quantifiers if and only if it is either an algebraically closed field, a finite field, or a 2 x 2 matrix ring over a finite prime field. Two features of this proof merit mentioning in this introduction. In [11] Rose and Woodrow developed the concept of ultrahomogeneity as distinct

www.csd.uwo.ca/∼moreno Abstract. Quantifier elimination (QE) over real closed fields has found numerous applications. Cylindrical algebraic decomposition (CAD) is one of the main tools for handling quantifier elimination of nonlinear input formulas. Despite of its worst case doubly exponential

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In this short expository note, we present a self-contained proof that algebraically closed fields of characteristic zero admit elimination of quantifiers over the elementary language of rings. We do this by fleshing out a method due to Muchnik 1 (and possibly Cohen 2). This note is the first part

. Additionally we find that the continuum hypothesis holds for classes of sets definable in monadic second-order logic over finitely branching trees, which is notable for not all of these classes are analytic. Our approach is based on Shelah's composition method and uses basic results from descriptive set

to any S-variety. We give a geometric proof of relative quantifier elimination for pseudo-finite fields, and we construct a morphism from the Grothendieck ring of the theory of pseudo-finite fields over S, to the tensor product of Q with the Grothendieck ring of constructible effective Chow motives

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an absolute constant c such that b(G) c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) 4, or G = U6(2) · 2, Gω = U4(3) · 22 and b(G) = 5. The proof is probabilistic, using bounds on fixed

, based on elimination of quantifiers, one used for solving unification and disunification problems, the other used to solve polynomials. This combination provides a procedure to solve RH-constraints in the context of HH with constraints.