analytic combinatorics in several variables (aimath, san jose 4-9.4.2022)

\(\def\Comp{\mathbb{C}}\def\Proj{\mathbb{P}}\def\Nat{\mathbb{N}}\)

This workshop is about complex analytic techniques usable in applications from classical combinatorial problems to asymptotic representation theory and cluster algebras. The scope is approximately what is covered by Pemantle-Wilson(-Melczer) book.

I plan to report on what is going on here (reporting is sporadic and idiosyncratic). Lectures are streamed.

Day One (all times PDT):

introductory lectures 1, 9:30am
  • 9:30 Robin – introduction to the area
    …big problem – are all D-finite (univariate) generating functions diagonals of rational ones?..
    …smaller problem – are there an algorithm detecting (local) irreducibility of a variety?
  • 10:45 Steve – algorithmic aspects
    …what is good software (does it exist?) to compute the flows, – say, gradient ones, – on submanifolds (algebraic is fine)?
    …the homology class of torus defining the coefficients into the linear combination of classes (tubes around) the unstable varieties of the critical points for the phase function on the variety. How to find the coefficients of this decomposition (in dimensions \(>2\))?
  • 11:35 Yuliy – amoebas and their uses
problem session 1, 2:30pm
  • Robin: Consider \(f(x_1,\ldots,x_d)\), \(p\) a singular point of \(f\). When locally, near \(p\) the variety defined by \(f\) decomposes into union of closed analytic varieties? Equivalently, when \(f\) factors in the ring of germs of analytic functions at \(p\)? We need an algorithm detecting (or refuting) that.
  • Joe: Consider a rational function \(R=P(x,y,z)/Q(x,y,z)\) and replace \(z^n\) by \(n!\). What can be said about the resulting generating function in \(x,y\)?
  • Mark: Consider algebraic generating function \(F\), the diagonal of a rational \(R(x,y)\). If coefficients of \(F\) are nonnegative rational, can we always find \(R\) with the same property?
  • Steve: If a linear function on the contour of a variety (the critical set for the Log mapping restricted to the variety of a Laurent polynomial) has a local minimum, is this a critical point?
  • Robin: \(Q(x_1,\ldots,x_d)\) defines a relation on \(R\subset V_Q\times \Comp\Proj^{d-1}\):
    \[(z,q)\in R\Leftrightarrow z \mbox{ is the critical point of phase defined by } q.\]
    Q1: If \(q\) is the diagonal vector, \(Q\) is symmetric multilinear, then the critical point on the diagonal is a root of \(Q(x,x,\ldots,x)=0\). What about other directions?
    Q2: (TBA)
  • Igor: \(\Nat\)-D-finite functions are defined as solutions of systems of linear ODEs with coefficients in \(\Nat\). What can be said about this class of functions?

Day Two (all times PDT):

introductory lectures 2, 9:15am
  • 9:15 Mark – mostly around this paper.
    Algebraic generating functions: need more work.
    Diagonals and rationalization of algebraic generating functions.
  • 10:00 Terence: toric varieties (corresponding to Newton polytope of a Laurent polynomial), and the behavior of the varieties at “infinities.”
problem session 2, 10:45am
  • Sheila: Consider the family of polynomials:
    \[F_2=t^x+ty; q_n=\frac12 (F_{2n}(t,x,y)+F_{2n}(t,x,-y));R_{2n}(x,y)=[t^{2n}]q_{2n}(t,x,y); \]\[F_{2n+2}t^{2n}R_{2n}(x(t^2x+ty+1,tx+y)-F_{2n}.\]
    Question: are the coefficients of \(q\) nonnegative? They describe certain components of the permutation action on the homologies of the even partitions lattice.
  • Robin: Can we rethink the problem of decomposition of the natural cycles on the variety coming from the Cauchy integral in terms of unstable cycles of critical points in terms of differential forms?

Day Three: research groups working

  • Rationalizing algebraic generating functions: what can be automated, and what can be still done by hand?
  • Boundary points of toric varieties (where the varieties \(Q=0\) compactify nicely) and the stationary points at infinity: what can be said or glimpsed?
  • \(\Nat\)-D-finite functions: what are they?
  • Manipulating amoebas

Day Four: mostly JMM session on ACSV

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