# Dimension of Ideals

Let $I$ be an ideal of the local polynomial ring $K[x_1, \ldots, x_n]_{\langle x_1, \ldots, x_n\rangle}$, where $K$ is a field. As for polynomial rings, the dimension of the ideal $I$ can be defined as the the maximum of the cardinalities of all the independent sets modulo $I$ (see Section [Dimension of Ideals](#gb-secdim) for details).

## `Dimension(I): RngMPolLoc -> RngIntElt, [ RngIntElt ]`

Given an ideal $I$ of a local polynomial ring $R$ defined over a field, return the dimension $d$ of $I$, together with a (sorted) sequence $U$ of integers of length $d$ such that the variables of $P$ corresponding to the integers of $U$ constitute a maximally independent set modulo $I$. If $I$ is the full local polynomial ring $R$, the dimension is defined to be $-1$, and the second return value is not set. The algorithm implemented is that given in [[Becker and Weispfenning, 1993](../../references.md#cite-beckerweispfenning), p. 449].