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The authors present deterministic algorithms with single exponential bit complexity bounds for computing the Chow forms and Hurwitz forms of projective varieties defined over integers.

Abstract

The paper focuses on the computational complexity of Chow forms and Hurwitz forms, which are fundamental objects in algebraic geometry and elimination theory.
Key highlights:
The authors provide the first bit complexity bounds for computing Chow forms, improving upon previous results that were in the arithmetic complexity model.
They develop a deterministic algorithm to compute the Chow form of a pure dimensional variety, with a single exponential bit complexity bound.
The algorithm is extended to compute Hurwitz forms, also with precise complexity estimates.
The authors further generalize their techniques to compute multiprojective Chow forms, taking into account the partition of variables into blocks.
Connections between multiprojective Chow/Hurwitz forms and matroid theory are explored, showing that the set of non-degenerate dimension vectors forms a polymatroid.
The authors' motivation comes from incidence geometry, where the computational complexity of Chow forms is crucial for addressing open problems.

Stats

The bitsize of the Chow form CFV is e
O(ndr-1τ), where d is the maximum degree of the defining polynomials, r is the dimension of the variety V, and τ is the bitsize of the coefficients of the defining polynomials.
The bit complexity of the algorithm to compute CFV is e
OB(m2m2+κ n r6nr(n-r)(ω+1)n (2d)2m2n+ωn2r+(ω+1)n (τ + n)), where ω is the exponent of matrix multiplication and κ is a small constant.

Quotes

"The Chow form is now recognized as a fundamental construction in algebraic geometry and it is particularly important in elimination theory."
"Our paper fills this gap. We also extend our algorithmic results to Hurwitz forms [53] and to the recent generalization of Chow forms for multiprojective varieties [46]."
"Contrary to the homogeneous case, multigraded Chow forms and Hurwitz form require a choice of a non-degenerate multidimension vector for the linear subspace, in a sense that is discussed in Section 4."

Deeper Inquiries

The algorithms presented in the paper can be applied to solve specific problems in incidence geometry by providing a systematic way to compute Chow and Hurwitz forms of projective varieties. In the context of incidence geometry, these forms play a crucial role in understanding the intersection patterns of linear subspaces with the given variety. By computing the Chow form, one can determine the algebraic representation of the variety, which is essential for studying its properties and relationships with other geometric objects. Additionally, the Hurwitz form helps in characterizing the non-transversal intersections of linear subspaces with the variety, providing valuable information about the geometry of the space.
Specifically, in the context of incidence geometry, these algorithms can be used to analyze incidence structures, study the incidence relations between geometric objects, and investigate the combinatorial properties of incidence configurations. By computing the Chow and Hurwitz forms, researchers can gain insights into the incidence patterns of linear subspaces with the projective variety, leading to a deeper understanding of the geometric relationships and structures involved in the incidence geometry problems.

The connections between multiprojective Chow and Hurwitz forms and matroid theory have significant implications for both algebraic geometry and combinatorics. Matroid theory provides a combinatorial framework for studying independence structures, and the relationship between multiprojective forms and matroids opens up new avenues for exploring the combinatorial properties of algebraic varieties.
The connection between multiprojective Chow and Hurwitz forms and matroid theory allows for the exploration of geometric objects through a combinatorial lens. By viewing the non-degenerate dimension vectors as elements of a polymatroid, researchers can analyze the combinatorial structure of these forms and derive insights into the underlying geometric properties. This connection can lead to new results in both algebraic geometry and combinatorics, providing a deeper understanding of the interplay between geometric structures and combinatorial properties.
Furthermore, the application of matroid theory to the study of multiprojective forms can potentially lead to new applications in other areas of mathematics, such as optimization, graph theory, and theoretical computer science. The rich combinatorial structure of matroids can offer new perspectives on the complexity and structure of algebraic varieties, opening up avenues for interdisciplinary research and cross-disciplinary collaborations.

There are several fundamental objects in algebraic geometry and computational algebra that could benefit from a similar complexity-theoretic analysis as the one provided for Chow and Hurwitz forms. Some of these objects include:
Resultants and Discriminants: Similar to the analysis of Chow and Hurwitz forms, studying the complexity of computing resultants and discriminants of polynomial systems can provide insights into the algebraic properties of varieties and their intersections. Understanding the bit complexity of these computations can lead to more efficient algorithms for solving polynomial equations and characterizing geometric objects.
Intersection Theory: Analyzing the complexity of intersection theory in algebraic geometry, such as computing intersection numbers, multiplicities, and intersection products, can enhance our understanding of the geometry of varieties and their relationships. By investigating the bit complexity of intersection computations, researchers can develop faster algorithms for studying the intersection properties of algebraic varieties.
Toric Varieties and Convex Geometry: Exploring the complexity of computations related to toric varieties, convex polytopes, and their connections to algebraic geometry can provide valuable insights into the geometric and combinatorial properties of these objects. Understanding the computational aspects of toric varieties can lead to advancements in computational geometry and optimization.
By conducting a complexity-theoretic analysis of these fundamental objects in algebraic geometry and computational algebra, researchers can improve the efficiency of computational algorithms, deepen their understanding of geometric structures, and pave the way for new applications in mathematics and related fields.

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