Irreducibility of Full Multivariate Partial-Twuality Polynomials
Abstract
For each nonempty finite simple graph, we study five full multivariate partial-twuality polynomials defined using matrix ranks and coranks over 𝔽2. Over an arbitrary coefficient field K, we give a necessary and sufficient criterion for nontrivial factorization in the Laurent polynomial ring by additive separation of the exponent function across a nontrivial vertex partition. For •∈{τ,δτ,τδ,δτδ}, the full multivariate partial-⟨•⟩ polynomial is irreducible in K[z±1,u,v] if and only if the graph is connected. For δ, we characterize the connected graphs whose polynomials are reducible by a condition on the parities of vertex degrees and of the numbers of common neighbors, and determine their irreducible factors. For each operation, the corresponding polynomial is irreducible in K[z,u,v] if and only if the graph is connected and at least one state has zero z-exponent. In particular, the full multivariate partial-⟨δ⟩ polynomial is irreducible in K[z,u,v] if and only if the graph is connected and bipartite.
Related
- Partial-Twuality Polynomials of Paired Matrices
- The Odlyzko-Poonen Conjecture on Irreducibility of Random Polynomials
- Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters
- Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density
- A proof of the irreducibility conjecture for Legendre polynomials
BibTeX
@misc{deng2026irreducibility,
title = {{Irreducibility of Full Multivariate Partial-Twuality Polynomials}},
author = {Qingying Deng and Chaoyang Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01445},
eprint = {2611.01445},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01445},
}
RIS
TY - UNPB
TI - Irreducibility of Full Multivariate Partial-Twuality Polynomials
AU - Qingying Deng
AU - Chaoyang Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01445
UR - https://arxiv.org/abs/2611.01445
AB - For each nonempty finite simple graph, we study five full multivariate partial-twuality polynomials defined using matrix ranks and coranks over 𝔽_2. Over an arbitrary coefficient field K, we give a necessary and sufficient criterion for nontrivial factorization in the Laurent polynomial ring by additive separation of the exponent function across a nontrivial vertex partition. For •∈{τ,δτ,τδ,δτδ}, the full multivariate partial-⟨•⟩ polynomial is irreducible in K[z^±1,u,v] if and only if the graph is connected. For δ, we characterize the connected graphs whose polynomials are reducible by a condition on the parities of vertex degrees and of the numbers of common neighbors, and determine their irreducible factors. For each operation, the corresponding polynomial is irreducible in K[z,u,v] if and only if the graph is connected and at least one state has zero z-exponent. In particular, the full multivariate partial-⟨δ⟩ polynomial is irreducible in K[z,u,v] if and only if the graph is connected and bipartite.
ER -