Uniform families of polynomial equations over a finite field and structures admitting an Euler characteristic of definable sets. (Q2766366)

The pigeonhole principle can be expressed by a uniform system of polynomial equations, by using variables \(x_{i,j}\). (\(x_{i,j}\) is \(1\) or \(0\) as pigeon \(i\) is in hole \(j\), or not.) The solvability depends only on whether the base set is finite or infinite, as the principle claims the non-existence of an injection into a proper subset. But when one considers a structure (rather than just a set), together with an Euler characteristic, the solvability situation is quite different, and the finite/infinite distinction is not decisive anymore. The main aim of this paper is the solvability criterion [Theorem 4.1.]. The Euler characteristic, here, is a cardinality function, and has little to do with cohomology. Given a structure \(M\) and a ring \(R\), it is a function \(\chi: \text{Def}(M)\to R\), with certain properties, like \(\chi(A\cup B)= \chi(A)+ \chi(B)\) provided \(A\) and \(B\) are disjoint. The author investigates its nature, existence, definability, and so forth. The `uniformity' of a polynomial system is defined by `definability', but an intuitive feeling is that the indices of variables have no structure like order. So, for example, the ``house sitting principle'' is not uniform. The motivation of this paper, the author states, lay in ideal membership proof systems like Nullstellen and Gröbner calculus, and he gives a result about degree lower bounds [Theorem 6.1.]. The author gives a number of examples, as usual.