[Paper Review] Bit-size estimates for triangular sets in positive dimension

Xavier Dahan, Abdulilah Kadri+1 more|arXiv (Cornell University)|Aug 20, 2010

Advanced Numerical Analysis Techniques14 references4 citations

TL;DR

This paper provides worst-case bit-size estimates for coefficients in triangular sets representing positive-dimensional algebraic sets over ℚ, extending prior work limited to zero-dimensional cases. By combining evaluation and interpolation techniques with height theory, it derives explicit upper bounds depending only on the degree and height of the underlying algebraic set, enabling theoretical analysis of modular algorithms and coefficient growth in polynomial system solving.

ABSTRACT

We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the underlying algebraic sets. We illustrate the use of these results in the context of a modular algorithm. This extends results by the first and last author, which were confined to the case of dimension 0. Our strategy is to get back to dimension 0 by evaluation and inter- polation techniques. Even though the main tool (height theory) remains the same, new difficulties arise to control the growth of the coefficients during the interpolation process.

Motivation & Objective

• To extend prior bit-size estimates for triangular sets from zero-dimensional to positive-dimensional algebraic sets over ℚ.
• To address the challenge of controlling coefficient growth during interpolation in positive-dimensional settings.
• To provide theoretical bounds on coefficient bit-sizes that are essential for analyzing probabilistic modular algorithms.
• To bridge the gap in coefficient size estimation for rational function fields ℚ(𝑌), where previous methods failed due to lack of Archimedean absolute values.
• To support the design and analysis of efficient algorithms for triangular decomposition in positive dimensions.

Proposed method

• Adapting evaluation and interpolation techniques to reduce positive-dimensional problems to zero-dimensional ones.
• Using height theory and arithmetic Nullstellensatz to bound coefficient sizes in the interpolated results.
• Applying the arithmetic Bézout theorem and degree/height estimates on resultants and algebraic sets to control growth during interpolation.
• Employing matrix minors and linear combinations to triangulate systems and bound degrees and heights of intermediate polynomials.
• Deriving explicit bounds via the arithmetic Nullstellensatz (Krick et al., 2001) and applying it to systems involving $1 - SH$, $J$, and $f_i$.
• Combining bounds on individual components ($A_2$, $A_3$) to derive global bit-size estimates for the final triangular set coefficients.

Experimental results

Research questions

• RQ1 What are the worst-case bit-size bounds for coefficients in triangular sets representing positive-dimensional algebraic sets over ℚ?
• RQ2 How can coefficient growth during interpolation be controlled when lifting zero-dimensional results to positive dimensions?
• RQ3 Can height theory be extended to provide bit-size estimates for rational function fields ℚ(𝑌) in positive-dimensional settings?
• RQ4 What role do the Chow form and resultant play in estimating the complexity of triangular representations?
• RQ5 How can these bounds be used to analyze the success probability of modular algorithms for triangular decomposition?

Key findings

• The paper establishes a bit-size bound of $ O((m+n)^2 n^5 d^{4n+4}(n h + (m+n)^2 /log(d))) $ for the coefficients in the triangular set, where $ d $ is the degree, $ h $ the height, and $ m,n $ the number of variables and equations.
• The bound for $ h(A_2) $, corresponding to the interpolation step, is derived using the arithmetic Nullstellensatz and degree/height estimates on resultants and algebraic sets.
• For $ A_3 $, the height is bounded by $ O(d^{4n}(m n h + m^2 n /log(d) + m(m+n)/log(m+n))) $, reflecting the complexity of coefficient reconstruction.
• The method successfully overcomes the limitation of prior work by handling the absence of Archimedean absolute values in $ \mathbb{Q}(\mathbf{Y}) $ through interpolation-based reduction.
• The derived bounds are explicit and applicable to modular algorithms, enabling theoretical analysis of success probabilities and coefficient growth.
• The results extend the applicability of height-based complexity analysis to positive-dimensional systems, filling a critical gap in algorithmic algebraic geometry.