The power-compositions determinant and its application to global optimization (Q2784357)

Let \(C(n,p)=\{(a_1, \ldots, a_p)\in \mathbb Z_{\geq 0}^p: \sum_i a_i=n \}.\) For \(a,b\in C(n,p)\) define \(a^b=a_1^{b_1}\ldots a_n^{b_n}.\) Order the \(N={n+p-1 \choose p-1}\) elements of \(C(n,p)\) lexicographically, say, and form the \(N\times N\) matrix \(M(n,p)=(a^b).\) The main theorem is the following factorization theorem for the determinant of \(M(n,p):\) NEWLINE\[NEWLINE\det(M(n,p))=\prod_{k=1}^{\min\{n,p\}} \left(n^{{n-1\choose k}} \prod_{i=1}^{n-k+1} i^{(n-i+1){n-i-1 \choose k-2}} \right)^{{p\choose k}}.NEWLINE\]NEWLINE While this interesting theorem's proof occupies eight pages, the abstract underlines as a simple consequence that the polynomials \(p_a(x)=(a_1x_1+\ldots+a_px_p)^n\) with \(a\in C(n,p)\) form a basis for the vector space of homogeneous polynomials in \(\mathbb R[x_1,\ldots, x_p]\) of degree \(n.\) NEWLINENEWLINENEWLINEHowever, this fact is known since approximately a century as Biermann's theorem and has an easy proof: see p.31 of \textit{B. Reznick} [Sums of even powers of real linear forms, Mem. Am. Math. Soc. 463 (1992; Zbl 0762.11019)]. It is known that every \(C^2\)-function on a compact subset of \(\mathbb R^p\) can be written as a difference of convex functions; to do this constructively is in general difficult. For homogeneous polynomials Biermann's theorem yields an easy solution.