Name
Abstract | ||
|---|---|---|
Let C(n,p) be the set of p-compositions of an integer n, i.e., the set of p-tuples $\alpha=(\alpha_1,\ldots,\alpha_p)$ of nonnegative integers such that $\alpha_1+\cdots+\alpha_p=n$. The main result of this paper is an explicit formula for the determinant of the matrix whose entries are $\alpha^{\beta}=\alpha_1^{\beta_1}\cdots\alpha_p^{\beta_p}$ where $\alpha,\beta\in C(n,p)$. The formula shows that the determinant is positive and has a nice factorization. As an application, it is shown that the polynomials $p_\alpha(x)=(\alpha_1x_1+\cdots+\alpha_px_p)^n$ with $\alpha\in C(n,p)$ form a basis of the vector space Hn[x1, . . . ,xp] of homogeneous polynomials of degree n in p variables. The result is of interest in the context of global optimization because it allows an explicit representation of polynomials as a difference of convex functions. |
Year | DOI | Venue |
|---|---|---|
2001 | 10.1137/S0895479800369141 | SIAM Journal on Matrix Analysis and Applications |
Keywords | Field | DocType |
composition,homogeneous polynomial,Vandermonde's determinant,difference of convex functions | Alpha (ethology),Integer,Vector space,Combinatorics,Matrix (mathematics),Mathematical analysis,Convex function,Homogeneous polynomial,Factorization,Vandermonde matrix,Mathematics | Journal |
Volume | Issue | ISSN |
23 | 2 | 0895-4798 |
Citations | PageRank | References |
2 | 0.55 | 0 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Josep M. Brunat | 1 | 42 | 5.52 |
| Antonio Montes | 2 | 205 | 19.68 |