split-primes-elekes-ronyai

name	split-primes-elekes-ronyai
description	Split primes and Elekes-Rónyai problem methodology for number theory counterexamples and arithmetic combinatorics
category	number-theory
tags	["number-theory","arithmetic-combinatorics","prime-numbers","expander-polynomials","subquadratic-growth"]

Split Primes and Elekes-Rónyai Problem

Description

Methodology for constructing counterexamples to the Elekes-Rónyai problem, demonstrating that certain polynomials which are neither additive nor multiplicative can still produce subquadratic expansion. Based on the technique showing |{x+y+(x-y)²: x,y∈A}| ≤ |A|^(2-c) for absolute constant c>0.

Activation Keywords

• Elekes-Ronyai
• split primes
• subquadratic expansion
• polynomial expansion
• arithmetic combinatorics
• sum-product phenomena
• 素数分裂
• 亚二次扩张
• 算术组合学

Core Concepts

The Elekes-Rónyai Problem

The classical Elekes-Rónyai theorem states that for any bivariate polynomial f(x,y) ∈ R[x,y] that is neither of the form g(h(x)+k(y)) nor g(h(x)·k(y)), the image set |f(A,A)| grows quadratically (|A|²) for large finite sets A.

The Counterexample Construction

The counterexample uses f(x,y) = x+y+(x-y)² which is neither additive nor multiplicative in the Elekes-Rónyai sense, yet produces subquadratic growth:

• |{x+y+(x-y)²: x,y∈A}| ≤ |A|^(2-c)
• For some absolute constant c > 0
• Achieved through careful construction of sets A ⊂ R

Connection to Split Primes

The construction relates to split primes in number fields, where the splitting behavior of primes can be used to construct sets with controlled expansion properties.

Usage Patterns

Pattern 1: Counterexample Construction

1. Identify polynomial f(x,y) that avoids both additive and multiplicative forms
2. Construct set A using split prime properties in number fields
3. Bound |f(A,A)| using arithmetic combinatorics techniques
4. Verify subquadratic growth rate

Pattern 2: Expansion Rate Analysis

1. For given polynomial f, determine if it falls in additive/multiplicative class
2. If not, check if counterexample construction applies
3. Calculate expansion exponent c using the methodology
4. Compare with theoretical bounds

Mathematical Framework

Key Ingredients

1. Split Prime Construction: Use algebraic number theory to construct sets A with controlled arithmetic properties
2. Expansion Bounds: Apply combinatorial techniques to bound |f(A,A)|
3. Counterexample Verification: Show f is neither additive nor multiplicative form

Applications

• Additive combinatorics research
• Sum-product phenomenon extensions
• Polynomial growth classification
• Number field arithmetic

Error Handling

Polynomial Classification

• If f(x,y) = g(h(x)+k(y)) or g(h(x)·k(y)), the Elekes-Rónyai theorem applies directly
• Must verify f does NOT decompose into these forms before claiming counterexample

Set Construction

• Set A must be carefully constructed — random sets typically achieve quadratic expansion
• Use split prime properties for controlled construction

References

• arXiv:2606.13619 — Split primes and the Elekes-Rónyai problem
• Elekes, Rónyai — Original theorem on polynomial expansion
• Solymosi — Survey on sum-product problems