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Efficient congruencing in ℤ[t] modulo t2

  • Trevor D. Wooley
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Abstract

We develop an analogue of nested efficient congruencing applicable in the ring ℤ[t]/t2ℤ[t] so as to estimate mean values related to that of Vinogradov. Let k be a natural number, and suppose that φj(u)∈ ℤ[t,u] (1≤ j≤ k) is a system of polynomials having Wronskian W(u;φ)=det (diφj(u)/dui)1≤ i,j≤ k with respect to u. Consider the number Υs,k(X;φ) of integral solutions of the system of equations ∑i=1s ( φj(ui+vit)−φj(us+i+vs+it)) ≡ 0 pmodt2 (1≤ j≤ k), with |ui|≤ X and |vi|≤ X (1≤ i≤ s). We show that when W(u;φ)not≡ 0pmodt and 1≤ s≤ k(k+1)/2, then Υs,k(X;φ)≪ X2s+ε. Our conclusions extend to number fields and function fields, as we outline. We describe several consequences of our results, including forthcoming work establishing the main conjecture for certain systems of equations not possessing translation-dilation invariant properties.

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Cite

BibTeXRIS
BibTeX
@misc{wooley2026efficient,
  title = {{Efficient congruencing in $\mathbb Z[t]$ modulo $t^2$}},
  author = {Trevor D. Wooley},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03487},
  eprint = {2611.03487},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03487},
}
RIS
TY  - UNPB
TI  - Efficient congruencing in ℤ[t] modulo t^2
AU  - Trevor D. Wooley
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03487
UR  - https://arxiv.org/abs/2611.03487
AB  - We develop an analogue of nested efficient congruencing applicable in the ring ℤ[t]/t^2ℤ[t] so as to estimate mean values related to that of Vinogradov. Let k be a natural number, and suppose that φ_j(u)∈ ℤ[t,u] (1≤ j≤ k) is a system of polynomials having Wronskian W(u;φ)=det (d^iφ_j(u)/du^i)_1≤ i,j≤ k with respect to u. Consider the number Υ_s,k(X;φ) of integral solutions of the system of equations ∑_i=1^s ( φ_j(u_i+v_it)−φ_j(u_s+i+v_s+it)) ≡ 0 pmodt^2 (1≤ j≤ k), with |u_i|≤ X and |v_i|≤ X (1≤ i≤ s). We show that when W(u;φ)not≡ 0pmodt and 1≤ s≤ k(k+1)/2, then Υ_s,k(X;φ)≪ X^2s+ε. Our conclusions extend to number fields and function fields, as we outline. We describe several consequences of our results, including forthcoming work establishing the main conjecture for certain systems of equations not possessing translation-dilation invariant properties.
ER  -