Applications of algebraic combinatorics to algebraic geometry (Q2243179)

Let \(k\) be a (perfect) field and \(K\) any extensions of \(K\). The common theme of this paper if the following one. Fix a degree \(d\) polynomial \(P: V\to k\) or a multilinear function \(P:V_1\times V_d\to k\) or a finite set of polynomials \(P_1,\dots,P_s\) of fixed degrees \(d_1,\dots ,d_s\) and their zero-locus \(X_{P_1,\dots ,P_s}\subset V\) over \(k\) and over \(K\). The dimensions of \(V\) or of \(V_1,\dots ,V_d\) are not fixed, only \(d\) or \(d_1,\dots ,d_s\). We have informations on \(X_P\) over \(K\) (e.g. Schmidt rank or slice rank) and we get informations over \(k\) multiplying the ``rank'' by a factor \(\epsilon _d\) (or \(\epsilon _{d_1,\dots ,d_k}\)) only depending of \(d\) not the dimensions of the vector space and the degree of the extensions of \(k\). If \(k\) is a finite field the key concept is the analytic rank (which involves an additive character \(k\to \mathbb {C}^\ast\)). The paper gives a number of new results in Algebraic Geometry and \(p\)-adic rings and outline their proofs from a proved theorem in the set-up of Algebraic combinatorics. I only state the names of some sections: irreducibility of the fibers; universality of high rank polynomial mappings; the universality for number fields; weakly polynomial functions; Nullstellsatz; \(p\)-adic bias rank.