With k an infinite field and τ1,τ2 endomorphisms of km, we provide a dimension bound on open locus determinantal scheme, under which, for general subspace V⊆km n≤m/2, v1,v2∈V have τ1(v1)=τ2(v2) only if v1=v2. Specializing to permutations composed by coordinate projections, obtain abstract proof the theorem [10].

We prove that semialgebraic sets of rectangular matrices a fixed rank, skew-symmetric rank and real symmetric whose eigenvalues have prescribed multiplicities are minimal submanifolds the space given size.

in this paper, we give some determinantal and permanental representations of generalized lucas polynomials, which are a general form of generalized bivariate lucas p-polynomials, ordinary lucas and perrin sequences etc., by using various hessenberg matrices. in addition, we show that determinant and permanent of these hessenberg matrices can be obtained by using combinations. then we show, the ...