Density of solutions for systems of forms

2025/07/15 by Amichai Lampert, Lampert, Amichai · 1 citation
Computer Science · Engineering · #11D72 #11G25 #11G35 #11P55 #14G05 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Structural Analysis and Optimization

paper · pdf · doi:10.48550/arxiv.2507.11514

openalex publication_date 2025/07/15 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28

Abstract

Let K be a field of characteristic zero over which every diagonal form in sufficiently many variables admits a nontrivial solution. For example, K may be a totally imaginary number field or a finite extension of a p-adic field. Suppose f₁,…,f_s are forms of degree d over K. Bik, Draisma and Snowden recently proved that there exists a constant B = B(d,s,K) such that the rational solutions to the system of equations f₁=…=f_s = 0 are Zariski dense, as long as the Birch rank of f₁,…,f_s is greater than B. We establish an effective bound for this constant, improving vastly on the astronomical bound coming from their proof. Our result has applications for surjectivity of polynomial maps and for the Hardy-Littlewood circle method.

Citations

• Strength and partition rank under limits and field extensions
• Strength is bounded linearly by Birch rank
• The geometry of polynomial representations in positive characteristic
• Two improvements in Brauer's theorem on forms
• Small Subalgebras of Polynomial Rings and Stillman's Conjecture

Cited by

• Polynomial Bounds for Birch's Theorem

Related

• Two improvements in Birch's theorem on forms
• Polynomial Bounds for Birch's Theorem
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• Strength is bounded linearly by Birch rank
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