Over a fixed finite field ${\bf F}_p$, families of
polynomial equations
$f_i(x_1, \dots, x_{n_N}) = 0$ for $i = 1, \dots, k_N$,
that are uniformly determined by a parameter $N$, are
considered. The notion of a uniform family is defined in
terms of first-order logic.
A notion of an abstract Euler characteristic is used to
give sense to a statement that the system has a solution
for infinite $N$, and a statement linking the solvability
of a linear system for infinite $N$ with its solvability
for finite $N$ is proved.
This characterisation is used to formulate a criterion
yielding 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.