← 研踪 PaperTrailarXiv:2611.05289 · v1
Some algebraic and geometric computations in CH via Chow forms
首次提交 2026-11-04 02:03
版本更新 2026-11-04 02:03
雷达观测 2026-11-04 23:39
摘要
The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms. We improve the state-of-the-art complexity bounds from PSPACE to CH, the counting hierarchy, for a number of problems related to equidimensional decomposition, including deciding (1) whether the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety vanishes at a given point, (2) whether a given monomial is in the support of the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety, (3) membership in a given equidimensional component of a given algebraic variety, and (4) membership in the closure of the image of an irreducible variety under a polynomial map. As an application of our algorithm to decide membership in the closure of the image of a variety under a polynomial map, we show that the approximate polynomial satisfiability problem can be decided in CH. We also show that the degree of the $k^{\text{th}}$ equidimensional component of a given projective variety can be computed in $\text{FP}^{\text{CH}}$. Our main technical result is the construction of a polylogtime-uniform family of constant-depth, exponential-size threshold circuits that compute the Chow forms of the equidimensional components of a given projective variety. We construct these circuits by combining the machinery of geometric resolutions together with the uniform constant-depth, exponential-size threshold circuits of Andrews, Garg, and Schost (FOCS 2026) for the multivariate resultant.
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v1 · 2026-11-04 02:03 · 最新观测版本
Some algebraic and geometric computations in CH via Chow forms
The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms. We improve the state-of-the-art complexity bounds from PSPACE to CH, the counting hierarchy, for a number of problems related to equidimensional decomposition, including deciding (1) whether the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety vanishes at a given point, (2) whether a given monomial is in the support of the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety, (3) membership in a given equidimensional component of a given algebraic variety, and (4) membership in the closure of the image of an irreducible variety under a polynomial map. As an application of our algorithm to decide membership in the closure of the image of a variety under a polynomial map, we show that the approximate polynomial satisfiability problem can be decided in CH. We also show that the degree of the $k^{\text{th}}$ equidimensional component of a given projective variety can be computed in $\text{FP}^{\text{CH}}$. Our main technical result is the construction of a polylogtime-uniform family of constant-depth, exponential-size threshold circuits that compute the Chow forms of the equidimensional components of a given projective variety. We construct these circuits by combining the machinery of geometric resolutions together with the uniform constant-depth, exponential-size threshold circuits of Andrews, Garg, and Schost (FOCS 2026) for the multivariate resultant.
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