Universality of monomial square forms
Abstract
A monomial square form is a polynomial F=∑i=1k Mi2, where M1,…,Mk are pairwise distinct squarefree monomials ≠1. We consider the problem of which of these forms are universal, i.e.\ represent every positive integer. We consider three classes of such forms:\\ (1) Forms M12+M22+M32+M42 (k=4),\\ (2) Forms M12+… +M52 (k=5) in at most 5 variables,\\ (3) Forms where the monomials are products of consecutive variables of a fixed length:\\ Pm,d(x)=∑i=1d(xixi+1⋯ xi+m−1)2, and its circular version\\ Cm,d(x)=∑i=1n(xixi+1⋯ xi+m−1)2 (n=m+d−1, xi+n=xi). We classify which of these are universal, possibly universal, or conditionally universal on a conjecture on the form x2+4y2+5z2. This ternary form appears repeatedly and seems to play a central role. For forms (1), we also show that a form is universal if and only if it represents 14, 15, 29 and 9591 (partially conditional on x2+4y2+5z2).
Related
- Systems of parameters consisting of linear forms for monomial ideal quotients
- Proofs of strong forms of the Bovdi--Geche universality conjecture and a complex analogue
- Tight universality of m-gonal forms with minimal criterion sets
- Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field
- Dynamical invariants of monomial correspondences
BibTeX
@misc{maubach2026universality,
title = {{Universality of monomial square forms}},
author = {Stefan Maubach},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03057},
eprint = {2611.03057},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03057},
}
RIS
TY - UNPB
TI - Universality of monomial square forms
AU - Stefan Maubach
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03057
UR - https://arxiv.org/abs/2611.03057
AB - A monomial square form is a polynomial F=∑_i=1^k M_i^2, where M_1,…,M_k are pairwise distinct squarefree monomials ≠1. We consider the problem of which of these forms are universal, i.e.\ represent every positive integer. We consider three classes of such forms:\\ (1) Forms M_1^2+M_2^2+M_3^2+M_4^2 (k=4),\\ (2) Forms M_1^2+… +M_5^2 (k=5) in at most 5 variables,\\ (3) Forms where the monomials are products of consecutive variables of a fixed length:\\ P_m,d(x)=∑_i=1^d(x_ix_i+1⋯ x_i+m−1)^2, and its circular version\\ C_m,d(x)=∑_i=1^n(x_ix_i+1⋯ x_i+m−1)^2 (n=m+d−1, x_i+n=x_i). We classify which of these are universal, possibly universal, or conditionally universal on a conjecture on the form x^2+4y^2+5z^2. This ternary form appears repeatedly and seems to play a central role. For forms (1), we also show that a form is universal if and only if it represents 14, 15, 29 and 9591 (partially conditional on x^2+4y^2+5z^2).
ER -