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.
For an integer $q\ge2$, a matroid is $q$-entropic if its rank function, multiplied by $\log q$, is the joint-entropy function of random variables on a $q$-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over $\F_4$. The corresponding statements for alphabet sizes two and thr...
Mohammad Hossein Kalantari, Shahram Khazaei· 0 citations
We prove that the magnitude of the tope graph of a simple oriented matroid is the specialization at $x=-1$ of the motivic zeta function of its underlying matroid. This refines the Las Vergnas--Zaslavsky theorem and defines magnitude for arbitrary matroids. For simple orientable matroids, we compute the order of the pol...
We prove that the Ehrhart $h^*$-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and $(4,a,a+29)$, for all sufficiently large integers $a$, h...
We study the $q$-matroids associated with nondegenerate $\F_{q^m}$-linear near-MRD codes, which we call near-uniform $q$-matroids. Using the correspondence between rank-metric codes and their associated $q$-systems, we determine explicitly their rank functions and characterize their cyclic flats. We show that near-unif...
The recently established duality between tensor, symmetric and exterior products of uniform matroids on the one hand, and abstract (bi-)rigidity matroids on the other hand, connects theorems and open questions from both areas, previously thought unrelated. In particular, a 1981 question of Mason asks if any two uniform...
We prove measure-rigidity results for regular polynomial endomorphisms of~$\mathbb{C}^k$. For maps of the same degree whose leading homogeneous parts differ by an invertible linear map on the target, equality of equilibrium measures is equivalent to postcomposition by an affine symmetry of the Julia set. We then show t...
Yu-Gang Zhang· 2 citations
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