Folded-Algebraic Matroids: Characteristic Rigidity and Almost-Entropic Separation
Abstract
We introduce folded-algebraic matroids. In such a representation, every matroid element is replaced by a finite tuple of algebraic quantities, and transcendence degree agrees with matroid rank after one uniform scaling. The resulting class contains both algebraic and folded-linear matroids and is contained in the class of almost-entropic matroids, whose rank functions are limits of scaled entropy functions. We prove that the latter containment is proper. Our main result concerns the classical rank-three matroids M(p) of Gordon. For every prime p, we show that M(p) has a folded-algebraic representation over a field K if and only if K has characteristic p. We then use a point-identification construction that preserves almost-entropicity to obtain a $13$-element rank-three $3$-connected matroid C2,3 that is almost entropic but not folded algebraic. Choosing a common element as dealer also yields a connected $12$-participant port with incompatible characteristic requirements. Finally, we record compact explicit witnesses and size bounds for several other separating regions among the representation classes.
Related
BibTeX
@misc{khazaei2026foldedalgebraic,
title = {{Folded-Algebraic Matroids: Characteristic Rigidity and Almost-Entropic Separation}},
author = {Shahram Khazaei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01664},
eprint = {2609.01664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01664},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Folded-Algebraic Matroids: Characteristic Rigidity and Almost-Entropic Separation
AU - Shahram Khazaei
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01664
UR - https://arxiv.org/abs/2609.01664
AB - We introduce folded-algebraic matroids. In such a representation, every matroid element is replaced by a finite tuple of algebraic quantities, and transcendence degree agrees with matroid rank after one uniform scaling. The resulting class contains both algebraic and folded-linear matroids and is contained in the class of almost-entropic matroids, whose rank functions are limits of scaled entropy functions. We prove that the latter containment is proper. Our main result concerns the classical rank-three matroids M(p) of Gordon. For every prime p, we show that M(p) has a folded-algebraic representation over a field K if and only if K has characteristic p. We then use a point-identification construction that preserves almost-entropicity to obtain a $13$-element rank-three $3$-connected matroid C_2,3 that is almost entropic but not folded algebraic. Choosing a common element as dealer also yields a connected $12$-participant port with incompatible characteristic requirements. Finally, we record compact explicit witnesses and size bounds for several other separating regions among the representation classes.
ER -