We study the critical points of Laplace eigenfunctions on polygonal domains with a focus second Neumann eigenfunction. show that if each convex quadrilaterals has no eigenfunction an interior point, then there exists quadrilateral unstable point. also point second-Neumann Lip-1 polygon orthogonal sides is acute vertex.

Let K be a field and (f1,…,fs,ϕ) multivariate polynomials in K[x1,…,xn] (with s<n) each invariant under the action of Sn, group permutations {1,…,n}. We consider problem computing critical points ϕ restricted to algebraic set V(f), where f=(f1,…,fs). This is same as at which f vanishes Jacobian matrix associated rank deficient, provided that this finite. exploit invariance properties input spli...