Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPolynomial factorization statistics and point configurations in $mathbb{R}^3$
318
0
0.0
(
0
)
Authors
Trevor Hyde
Ask ChatGPT about the research
No Arabic abstract
We use generating functions to relate the expected values of polynomial factorization statistics over $mathbb{F}_q$ to the cohomology of ordered configurations in $mathbb{R}^3$ as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.
rate research
Read More
A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.
We prove an estimate for spherical functions $phi_lambda(a)$ on $mathrm{SL}(3,mathbb{R})$, establishing uniform decay in the spectral parameter $lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $mathrm{A}$. In the case of $mathrm{SL}(3,mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $lambda$ and $a$ vary.
Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $mathbb{F}_q[x_1, x_2, ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined rational function $M_{d,infty}(q)$. Furthermore we show that the limit $M_{d,infty}(q)$ is related to the classic necklace polynomial $M_{d,1}(q)$ by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of symmetric group representations as a consequence of liminal reciprocity.
We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group $G$ over a nonarchimedean local field $F$. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case $G=mathrm{GL}_r$ we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here `solvable means that the models have a family of Yang-Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang-Baxter equations come from a Drinfeld twist of the quantum group $U_q(widehat{mathfrak{gl}}(r|1))$, which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.
In the following article we discuss Delaunay triangulations for a point cloud on an embedded surface in $mathbb{R}^3$. We give sufficient conditions on the point cloud to show that the diagonal switch algorithm finds an embedded Delaunay triangulation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
