UNIFORM FAMILIES OF POLYNOMIAL EQUATIONS OVER A FINITE FIELD AND STRUCTURES ADMITTING AN EULER CHARACTERISTIC OF DEFINABLE SETS

Abstract

Over a fixed finite field ${\bf F}_p$, families ofpolynomial equations$f_i(x_1, \dots, x_{n_N}) = 0$ for $i = 1, \dots, k_N$,that are uniformly determined by a parameter $N$, areconsidered. The notion of a uniform family is defined interms of first-order logic.A notion of an abstract Euler characteristic is used togive sense to a statement that the system has a solutionfor infinite $N$, and a statement linking the solvabilityof a linear system for infinite $N$ with its solvabilityfor finite $N$ is proved.This characterisation is used to formulate a criterionyielding degree lower bounds for various ideal membership proof systems (for example,Nullstellensatz and the polynomial calculus).Further, several results about Euler structures(structures with an abstract Euler characteristic) are proved, and the case of fields, in particular,is investigated more closely. 1991 Mathematics Subject Classification: primary 03F20, 12L12, 15A06;secondary 03C99, 12E12, 68Q15, 13L05.