A Characterization of Zero-Dimensional Annihilators of Tableaus Original Research Article

In the theory of linear recursive sequences over a fieldKthere is a natural action of the polynomial ringK[x] on sequences (the end-off left shift), and every ideal ofK[x] is the annihilator with respect to this action of some sequences. This is a direct consequence of the fact thatK[x] is a principal ideal domain. In dimensionn ≥ 2 the analogue of a sequence is atableau, and there is an analogous action ofK[x1, x2,…,xn] on tableaus. However, a given ideal ofK[X] = K[x1, x2,…,xn] is not necessarily the annihilator of some tableau. We will give an example in Section 2. Our main result is to characterize an important subset of those ideals which are annihilators. More specifically, in the theory of linear recursive sequences, an idealIofK[x] and a finite set of initial conditions (the initial fill) completely determine a sequence annihilated byI. This is a consequence of the fact thatK[x]/Iis a finite-dimensionalK-vector space. In order for a tableau to be completely determined by an idealIand a finite set of initial conditions, we must have thatK[X]/Iis a finite-dimensionalK-vector space, i.e., the only primes containingIare maximal, i.e.,Iis a 0-dimensional ideal. In particular, ifmis maximal andIism-primary thenIis 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 isGorenstein.