XZ Conjecture for SU(N): convex-hull criterion from vanishing torus integrals

Prove that for any admissible function f: [0,1]^{N(N−1)/2} × T^{N(N+1)/2−1} → C, if for all positive integers P the integral ∫_{[0,1]^{N(N−1)/2}} ∫_{T^{N(N+1)/2−1}} (f(x,z))^P · ˜J_{SU(N)}(x) · (dz_1/z_1) … (dz_{N(N+1)/2−1}/z_{N(N+1)/2−1}) dx_1 … dx_{N(N−1)/2} equals 0, then the zero vector does not lie in the convex hull of Sp(f), where Sp(f) denotes the set of multi-indices m ∈ Z^{N(N+1)/2−1} for which the Fourier–Laurent coefficient c_m(x) in the expansion f(x,z)=∑_{m} c_m(x) z^m is nonzero and ˜J_{SU(N)}(x)=2^{N−1}C_N ∏_{j=1}^{N−1} x_j with C_N a constant.

Background

The paper develops a refined Euler angles parameterization and integration formula for SU(N), enabling the reduction of the Mathieu Conjecture to an abelian-style statement over products of intervals and tori without introducing N-th roots. Using this framework, the author formulates an explicit criterion involving integrals of powers of admissible functions against a Jacobian factor ˜J_{SU(N)}.

This conjecture serves as the central bridge to proving the Mathieu Conjecture for SU(N): the subsequent theorem shows that, assuming this conjecture holds, the Mathieu Conjecture follows. The admissible functions are those that admit finite Fourier–Laurent expansions in the torus variables with polynomial dependence on x and √(1−x^2), and Sp(f) is the set of exponent multi-indices of nonzero terms.