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.