A characterization of zero-dimensional annihilators of tableaus
In the theory of linear recursive sequences over a field \(K\) there is a natural action of the polynomial ring \(K[x]\) on sequences (the end-off left shift), and every ideal of \(K[x]\) is the annihilator with respect to this action of some sequence \(s\). This is a direct consequence of the fact that \(K[x]\) is a principal ideal domain. In dimension \(n\geq 2\) the analogue of a sequence is a tableau, and there is an analogous action of \(K[x_1,x_2,\dots,x_n]\) on tableaux. However, a given ideal of \(K[X]=K[x_1,x_2,\dots,x_n]\) is not necessarily the annihilator of some tableau. Our main result is to characterize an important subset of those ideals which are annihilators. In order for a tableau to be completely determined by an ideal \(I\) and a finite set of initial conditions, we must have that \(K[X]/I\) is a finite-dimensional \(K\)-vector space, i.e., the only primes containing \(I\) are maximal, i.e., \(I\) is a 0-dimensional ideal. In particular, if \(m\) is maximal and \(I\) is \(m\)-primary then \(I\) is 0-dimensional. Using methods of primary decomposition, we will be able to reduce to this case. Following a suggestion of B. Sturmfels, we show that a 0-dimensional ideal is the annihilator of some tableau if and only if it is Gorenstein.
scientific article; zbMATH DE number 1051587