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

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$ of affine or projective space. The bounds depend only on $\mathrm{card}(\mathcal P)$ and the degrees of the polynomials in $\mathcal P$, together with $\mathrm{deg}(V)$ and $\dim(V)$, and not on the dimension of the space in which $V$ is embedded. This feature is particularly useful when $V$ has small intrinsic dimension but is presented in a very high-dimensional ambient space. We describe several applications. First, we extend existing results on bounding the $\varepsilon$-entropy of real algebraic varieties. Second, we derive lower bounds (in terms of the number of connected components) for membership testing in semi-algebraic sets in the algebraic computation tree model. Finally, motivated by quantum complexity theory, we introduce additive and multiplicative notions of \emph{relative rank} in finite-dimensional vector spaces and algebras with respect to a fixed algebraic subset, generalizing the classical notion of tensor rank. We prove a general lower bound on the maximum relative rank of finite subsets with respect to algebraic sets of bounded degree and dimension that is again independent of the ambient dimension. As an illustration, we obtain a quantum analogue of Shannon's classical lower bound: almost all Boolean functions require classical circuits of size $Ω(2^n/n)$, even in the presence of a quantum oracle specified by an algebraic subset of fixed degree and dimension.