Variable Extension of Ideals

Often one wishes to introduce new variables temporarily to a polynomial ring. Magma allows one to do this by use of the VariableExtension function, and also to restrict again to the original ring with elimination performed automatically.

VariableExtension(I, k, b): RngMPol, RngIntElt, BoolElt -> RngMPol, Map

VariableExtension(I, k, b, order): RngMPol, RngIntElt, BoolElt, ... -> RngMPol, Map

Given an ideal \(I\) of the polynomial ring \(P = R[x_1, \ldots, x_n]\), create a polynomial ring \(Q\) as a \(k\)-variable extension of \(P\), the ideal \(J\) of \(Q\) corresponding to \(I\), and the embedding map \(f: P \rightarrow Q\), and return \(J\) and \(f\).

If the argument \(b\) (standing for “before”) is true, the \(k\) variables are inserted before the current variables of \(P\), so \(Q\) is defined to be \(R[y_1, \ldots, y_k, x_1, \ldots, x_n]\) and \(f\) maps \(P.i\) to \(Q.(k+i)\) (so the \(x_i\) variables of \(P\) are mapped to the \(x_i\) variables of \(Q\)).

If the argument \(b\) is false, the \(k\) variables are inserted after the current variables of \(P\), so \(Q\) is defined to be \(R[x_1, \ldots, x_n, y_1, \ldots, y_k]\) and \(f\) maps \(P.i\) to \(Q.i\) (so the \(x_i\) variables of \(P\) are mapped to the \(x_i\) variables of \(Q\)).

If the argument order is given, then \(Q\) is constructed with the specified order; otherwise, the grevlex order is used for \(Q\) by default. See the section on monomial orders (Section Representation and Monomial Orders) for the valid values for the argument order.

The image under \(f\) of a polynomial of \(P\) is the corresponding polynomial of \(Q\), while the image under \(f\) of an ideal of \(P\) is the corresponding ideal of \(Q\). The inverse image under \(f\) of a polynomial of \(Q\) is only defined if none of the extension variables of \(Q\) occur in that polynomial, in which case the inverse image is just the restriction back to \(P\), while the inverse image under \(f\) of an ideal \(H\) of \(Q\) is always defined and is the restriction back to \(P\) of the elimination ideal \(H \cap R[x_1, \ldots, x_n]\).