Bounds on the realizations of zero-nonzero patterns and sign conditions of polynomials restricted to varieties and applications

Abstract: We prove upper bounds which are independent of the dimension of the ambient space, on the number of realizable zero-nonzero patterns as well as sign conditions (when the field of coefficients is ordered) of a finite set of polynomials $\mathcal{P}$ restricted to some algebraic subset $V$ of an affine or projective space. Our bounds depend on the number and the degrees of the polynomials in $\mathcal{P}$, as well as the degree of $V$ and the dimension of $V$, but are independent of the dimension of the space in which $V$ is embedded. This last feature of our bounds is useful in situations where the ambient dimension is much larger than $\mathrm{dim} V$. We give several applications of our results. We generalize existing results on bounding the speeds of algebraically defined classes of graphs, bounding the $\varepsilon$-entropy of real varieties, as well as lower bounds in terms of the number of connected components for testing membership in semi-algebraic sets using algebraic computation trees. Motivated by quantum complexity theory we introduce a notion of relative rank (additive as well as multiplicative) in finite dimensional vector spaces and algebras relative to a fixed algebraic subset in the vector space or algebra -- which generalizes the classical definition of ranks of tensors. We prove a very general lower bound on the maximum relative rank of finite subsets relative to algebraic subsets of bounded degree and dimension which is independent of the dimension of the vector space or algebra.