A Solution to Iima--Yoshino Problem 2.3
Abstract
Iima and Yoshino asked for an ideal I in S=k[x1,x2,…], with deg xi=i, and a monomial order such that S/I≅ k[xi:i≡±1pmod5], in(I)=(xi2,xixi+1:i≥1). We construct such an ideal and monomial order over every field k of characteristic different from $5$ containing an element c with c2+c=1. The ideal has an explicit infinite homogeneous reduced Gröbner basis. A five-periodic syzygy derived from a pentagon identity proves that all basis relations belong to I and supplies standard representations for the non-coprime critical pairs. Triangular elimination establishes the graded quotient isomorphism. Together, the quotient and initial ideal descriptions yield the partition form of the first Rogers-Ramanujan identity. In each weighted degree, a perfect matching in the support of the normal-form matrix gives a bijection between the two partition classes. We formalize the complex specialization in Lean 4 using Mathlib and our set-based theory of infinite Gröbner bases, including the reduced basis, the graded quotient isomorphism, and the partition-matching theorem.
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BibTeX
@misc{guo2026solution,
title = {{A Solution to Iima--Yoshino Problem 2.3}},
author = {Junyu Guo and Hao Shen and Junqi Liu and Lihong Zhi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06477},
eprint = {2609.06477},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06477},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A Solution to Iima--Yoshino Problem 2.3
AU - Junyu Guo
AU - Hao Shen
AU - Junqi Liu
AU - Lihong Zhi
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06477
UR - https://arxiv.org/abs/2609.06477
AB - Iima and Yoshino asked for an ideal I in S=k[x_1,x_2,…], with deg x_i=i, and a monomial order such that S/I≅ k[x_i:i≡±1pmod5], in(I)=(x_i^2,x_ix_i+1:i≥1). We construct such an ideal and monomial order over every field k of characteristic different from $5$ containing an element c with c^2+c=1. The ideal has an explicit infinite homogeneous reduced Gröbner basis. A five-periodic syzygy derived from a pentagon identity proves that all basis relations belong to I and supplies standard representations for the non-coprime critical pairs. Triangular elimination establishes the graded quotient isomorphism. Together, the quotient and initial ideal descriptions yield the partition form of the first Rogers-Ramanujan identity. In each weighted degree, a perfect matching in the support of the normal-form matrix gives a bijection between the two partition classes. We formalize the complex specialization in Lean 4 using Mathlib and our set-based theory of infinite Gröbner bases, including the reduced basis, the graded quotient isomorphism, and the partition-matching theorem.
ER -