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

Saugata Basu, Laxmi Parida

TL;DR

This work derives ambient-dimension-free bounds on the realizations of zero-nonzero patterns and sign conditions for polynomials restricted to algebraic varieties, depending only on the number and degrees of the polynomials and the intrinsic degree and dimension of the variety. It differentiates algebraic and real-analytic techniques, employing generalized Bezout bounds and infinitesimal perturbations with projections to reduce to lower-dimensional real algebraic sets, then leverages Oleǐnik–Petrovskiĭ, Thom–Smith, and Lefschetz duality to bound Betti sums. The paper then applies these bounds to derive ${\varepsilon}$-entropy bounds for real varieties, improved lower bounds for algebraic computation trees, and a quantum-analogue of Shannon’s lower bound via relative ranks with respect to fixed algebraic subsets, yielding dimension-independent results. Collectively, these results contribute intrinsic geometric control over combinatorial and computational questions in real and complex algebraic geometry and quantum-information-inspired models. The framework enables robust, ambient-dimension-resilient analyses in discrete geometry, incidence geometry, and quantum circuit complexity with algebraic or semi-algebraic structures.

Abstract

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.

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

TL;DR

This work derives ambient-dimension-free bounds on the realizations of zero-nonzero patterns and sign conditions for polynomials restricted to algebraic varieties, depending only on the number and degrees of the polynomials and the intrinsic degree and dimension of the variety. It differentiates algebraic and real-analytic techniques, employing generalized Bezout bounds and infinitesimal perturbations with projections to reduce to lower-dimensional real algebraic sets, then leverages Oleǐnik–Petrovskiĭ, Thom–Smith, and Lefschetz duality to bound Betti sums. The paper then applies these bounds to derive -entropy bounds for real varieties, improved lower bounds for algebraic computation trees, and a quantum-analogue of Shannon’s lower bound via relative ranks with respect to fixed algebraic subsets, yielding dimension-independent results. Collectively, these results contribute intrinsic geometric control over combinatorial and computational questions in real and complex algebraic geometry and quantum-information-inspired models. The framework enables robust, ambient-dimension-resilient analyses in discrete geometry, incidence geometry, and quantum circuit complexity with algebraic or semi-algebraic structures.

Abstract

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 restricted to an algebraic subset of affine or projective space. The bounds depend only on and the degrees of the polynomials in , together with and , and not on the dimension of the space in which is embedded. This feature is particularly useful when 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 -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 , even in the presence of a quantum oracle specified by an algebraic subset of fixed degree and dimension.

Paper Structure

This paper contains 42 sections, 53 theorems, 204 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and prior work
  3. Basic definitions and notation
  4. Summary of main contributions
  5. Methods
  6. Tightness
  7. Applications
  8. Upper bounds on zero-nonzero patterns and sign conditions
  9. Bound on the number of realizable zero-nonzero patterns
  10. Bound on the number of realizable sign conditions
  11. Sum of Betti numbers of realizations of sign conditions
  12. Applications
  13. Bound on the ${\varepsilon}$-entropy of real varieties
  14. Lower bounds for algebraic computation trees
  15. Proofs
  16. ...and 27 more sections

Key Result

Theorem A

RBG01Jeronimo-Sabia Let $k$ be a field and $\mathcal{P} \subset k[X_1,\ldots,X_N]_{\leq d}$ be a finite set of polynomials of degree at most $d$. Then, the number of realizable zero-nonzero patterns on $\mathcal{P}$ (i.e. elements of $\{0,1\}^{\mathcal{P}}$) is bounded by $(O(s d))^N$, where $s = \m

Figures (1)

  • Figure 1: Schematic for the example with $p=1$ and $N=2$: $V(\mathrm{R})$ consists of $D^2$ disjoint ovals $V_{i,j}$, and ${\rm Z}(\mathcal{P},\mathrm{R}^2)$ is a collection of vertical lines $X_1=\delta_{i,j}$ intersecting each oval $V_{0,j}$ in $2sd$ points.

Theorems & Definitions (136)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Remark 1
  • Remark 2
  • Theorem F
  • Remark 3
  • Remark 4
  • ...and 126 more