Given a finite preorder $\tau$, we study how its order relation determines the $h^*$-polynomial of the associated polar preorder polytope. Let $\mathcal R_{\tau}^{\vee}$ be the convex hull of the negative standard basis vectors and the indicator vectors of nonempty order ideals, and write $h_\tau^*(t)=h^*(\mathcal R_{\tau}^{\vee},t)$. Athanasiadis and Chapoton conjectured that this enumerative invariant takes ordinal sums to products: $$h_{\tau_1\oplus\tau_2}^*(t)=h_{\tau_1}^*(t)\,h_{\tau_2}^*(t).$$ We prove the conjecture for arbitrary finite preorders, including those with nontrivial equivalence classes. More precisely, we construct a positively homogeneous piecewise unimodular shear from the free sum $\mathcal R_{\tau_1}^{\vee}\oplus\mathcal R_{\tau_2}^{\vee}$ onto $\mathcal R_{\tau_1\oplus\tau_2}^{\vee}$, giving lattice-point bijections at every nonnegative real dilation scale. The key is a formula for the Minkowski functional in terms of the least nonnegative order-reversing majorant: after the shear, the functional becomes the sum of the two factor functionals. Together with the Ehrhart product formula for free sums, this proves the conjecture and the corresponding product formula for normalized volume.
Let $A=\mathbb{F}_q[T]$ and $K=\mathbb{F}_q(T)$, and let $\phi$ be a Drinfeld $A$-module over $K$ of rank $r\geq 2$. We study the reduction of $\phi$ at primes $\mathfrak{p} \in A$ lying in a prescribed arithmetic progression, for which $\phi(\mathbb{F}_{\mathfrak{p}})$ has at most $r-1$ cyclic components. We obtain as...
Prassanna Nand Jha, Arnab Saha, A. Vatwani· 0 citations
Let $A=\{0=a_0<a_1<\dots<a_{k-1}=d\}$ be integers with greatest common divisor $1$, let $C_A$ be the projective monomial curve they define, and let $\mathrm{reg}(A)$ be the Castelnuovo-Mumford regularity of its coordinate ring. Gimenez and Gonz\'alez-S\'anchez proved $\mathrm{reg}(A)\le\sigma(A)+1$ for the sumsets regu...
We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on $L^\alpha$-flat polynomials. The theorem asserts that no $L^\alpha$-flat polynomials exist within the class of trigonometric polynomials of the form$$\sum_{m=1}^{n} a_m \cos(m\theta+\alpha_m), \qquad n \in \mathbb{N}^*,$$satisfying$$k\, n^...
Let $A=(a_{ij})$ be an $n\times n$ real matrix and let $\lambda$ be a partition of $n$. Let $\phi^\lambda$ be the class function dual to the Young permutation character, and let $$ \phi^\lambda[A] = \sum_{\sigma\in\mathfrak{S}_n}\phi^\lambda(\sigma) \prod_{i=1}^{n}a_{i\sigma(i)} $$ be the corresponding monomial immanan...
Let $(\mathcal{M},\tau)$ be a tracial von Neumann algebra, $T:L_1(\mathcal{M},\tau)\to L_1(\mathcal{M},\tau)$ be induced by a normal $\tau$-preserving $*$-automorphism of $\mathcal{M}$, and $x\in L_1(\mathcal{M},\tau)$. We will prove the bilaterally almost uniform convergence of averages of the form $$\frac{1}{n}\sum_{...
We prove a Poincar\'e--Birkhoff--Witt theorem for the distinguished affine super Yangian $Y=Y_{1/2,1/2}(\widehat{\mathfrak{sl}}(m|n))$, where $m,n\geq2$ and $m\neq n$. With respect to the current filtration, there is a canonical isomorphism \[ \operatorname{gr}_F Y\cong U\!\left(\operatorname{uce}\!\left( \mathfrak{sl}...
V. Volkov, V. Stukopin· 0 citations
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