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In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space - in particular, to every finite simple connected graph $G$ - and showed it has several attractive properties. First, it is multi-affine and real-stable (leading to a hitherto unstudied delta-matroid for each graph $G$). Second, the polynomial specializes to (a transform of) the characteristic polynomial $chi_{D_G}$ of the distance matrix $D_G$; as well as recovers the entire graph, where $chi_{D_G}$ cannot do so. Third, the polynomial encodes the determinants of a family of graphs formed from $G$, called the blowups of $G$. In this short note, we exhibit the applicability of these tools and techniques to other graph-matrices and their characteristic polynomials. As a particular case, we will see that the adjacency characteristic polynomial $chi_{A_G}$ is in fact the shadow of a richer multivariate blowup-polynomial, which is similarly multi-affine and real-stable. Moreover, this polynomial encodes not only the aforementioned three properties, but also yields additional information for specific families of graphs.
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To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X({ n_x : x in X })$. This is obtained from the blowup $X[{bf n}]$ - which contains $n_x$ copies of each point $x$ - by computing the determinant of the distance matrix of $X[{bf n}]$ and removing an exponential factor. We prove that as a function of the sizes $n_x$, $p_X({bf n})$ is a polynomial, is multi-affine, and is real-stable. This naturally associates a delta-matroid to each metric space $X$ (and another delta-matroid to every tree), which also seem to be hitherto unexplored. We moreover show that the homogenization at $-1$ of $p_X({bf n})$ is Lorentzian (or strongly/completely log-concave), if and only if the normalization of $p_X(-{bf n})$ is strongly Rayleigh, if and only if a modification of the distance matrix of $X$ is positive semidefinite. We next specialize to the case of $X = G$ a connected unweighted graph - so $p_G$ is partially symmetric in ${ n_v : v in V(G) }$ - and show two further results: (a) We show that the univariate specialization $u_G(x) := p_G(x,dots,x)$ is a transform of the characteristic polynomial of the distance matrix $D_G$; this connects the blowup-polynomial of $G$ to the well-studied distance spectrum of $G$. (b) We show that the polynomial $p_G$ is indeed a graph invariant, in that $p_G$ and its symmetries recover the graph $G$ and its isometries, respectively.
The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivaria
Recently, Nunge studied Eulerian polynomials on segmented permutations, namely emph{generalized Eulerian polynomials}, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eulerian polynomials and hence confirm Nunges conjecture. Our proof is based on Brandens stable multivariate Eulerian polynomials. By acting on Brandens polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunges conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.
We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for the expansion of a polynomial in terms of m-polynomials. We conclude this article by an implementation in MATHEMATICA of m-polynomials and the results obtained for them.
A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenbergs work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.
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