Über das assoziierte Ideal zyklischer Polytope (Q789451)

A ''cyclic polytope'' of type C(n,d) is by definition the convex hull of n distinct points of the image of the so called moment curve M: \({\mathbb{R}}\to {\mathbb{R}}^ d\) taking t to \((t,t^ 2,t^ 3,...,t^ d)\); it is assumed that \(n>d>0\). A cyclic polytope has the special property that each of its (d-1)-dimensional (bounding) faces is the convex hull of d affinely independent vertices \(y_ 1,y_ 2,...,y_ d\). It follows that each subset of \(\{y_ 1,...,y_ d\}\) generates a face of C(n,d) and hence the family of subsets of \(\{x_ 1,...,x_ n\}\) which span a face of C(n,d), is a simplicial complex \(\Delta\) (n,d). Next, if \(\Delta\) is a simplicial complex with vertex set \(\{x_ 1,...,x_ n\}\) and K is a field, let \(K[x_ 1,...,x_ n]\) be the polynomial ring on \(\{x_ 1,...,x_ n\}\) regarded as a set of commuting indeterminates, and consider the homogeneous ideal \(I_{\Delta}\) of \(K[x_ 1,...,x_ n]\) generated by all square-free monomials \(x_{i_ 1}...x_{i_ s}\) such that \(\{i_ 1,i_ 2,...,i_ s\}\) is not a simplex of \(\Delta\). If this construction is carried out for the simplicial complex \(\Delta\) (n,d) there results a quotient ring \(K[x_ 1,...,x_ n]/I_{\Delta(n,d)}\) which is a Gorenstein ring. Now the simplicial complex \(\Delta\) (n,d) can be described explicitly, but the description is a bit involved. So it comes as a surprise that a set of generators of \(I_{\Delta(n,d)}\) can be specified quite simply; indeed, the main result of the article under review states: Theorem. (i) If d is even, \(I_{\Delta(n,d)}\) is generated by all monomials \(x_{i_ 0}...x_{i_{(d/2)}}\) of degree \((d/2)+1\) with \(1\leq i_ 0,\quad i_ 0+1<i_ 1,\quad i_ 1+1<i_ 2,...,i_{(d/2)- 1}+1<i_{(d/2)}\leq n,\) and \(1<i_ 0\) or \(i_{(d/2)}<n\). (ii) If d is odd, \(I_{\Delta(n,d)}\) is generated by all monomials \(x_{i_ 0}...x_{i_{(d-1)/2}}\) of degree \((d+1)/2\) with \(1<i_ 0,\quad i_ 0+1<i_ 1,...,i_{(d-3)/2}+1<i_{(d-1)/2}<n\) together with all monomials \(x_ 1x_{i_ 1}...x_{i_{(d-1)/2}}x_ n\) of degree \((d+3)/2\) with \(2<i_ 1,\quad i_ 1+1<i_ 2,...,i_{(d- 3)/2}+1<i_{(d-1)/2}<n-1.\) Reviewer's remark: Good sources for background information on the connection between the theory of convex polytopes and commutative algebra are the article of \textit{R. P. Stanley} in Studies Appl. Math. 54, 135-142 (1975; Zbl 0308.52009) and the references given there. [See also the author's paper in J. Algebra 86, 272-281 (1984; Zbl 0533.13003).]