Naslund’s polynomial square-difference-free bound over F₃
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For polynomial sets over F₃ with no nonzero square differences, determine the largest size at a given degree cutoff. The counterexample target lifts a ten-word code in F₃⁴ to sets of size 810^e below degree 8e and 27·810^e below degree 8e+4.
Why it matters and what counts as progress
Why it matters: It tests Naslund’s proposed bound at q=3,k=2 and improves the growth-exponent lower bound to 16/21.
What counts as progress: Larger square-difference-free polynomial sets or sharper exponent bounds, with exact finite-field and degree restrictions.
How the problem developed
- 2023 · Bound and conjecture [1]
Eric Naslund Constructed square-difference-free subsets of F_q[T] of size q^(3n/4) for odd q and 4 dividing n, and conjectured this bound was tight.
J. D. Jones, with Claude, GPT and Grok agents contributing to construction and proof work Disproved the conjectured bound for q=3 at every n divisible by 4 with n ≥ 8; the formalized n=8 construction has 810 elements versus the conjectured 729.
AI contribution
AI-assisted construction refutes a Naslund bound over F₃
A ten-word code in F₃⁴ and an eight-degree lift give square-difference-free polynomial sets of size 810ᵉ below degree 8e and 27·810ᵉ below degree 8e+4. They refute Naslund’s Conjecture 13 at q=3, k=2 for every n divisible by four with n≥8, and imply a growth-exponent lower bound of 16/21.
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