Extension and Contraction of Ideals

Magma allows the extension to and contraction from the ring of quotients of an ideal, defined over a field, with respect to certain variables. See [Becker and Weispfenning, 1993, pp. 54--58 and 388--397] for the relevant definitions and theory.

Extension(I, U): RngMPol, [ RngIntElt ] -> RngMPol, Map

Given an ideal \(I\) of the polynomial ring \(P = K[x_1, \ldots, x_n]\), where \(K\) is a field, together with a sequence \(U\) of integers each between 1 and \(n\), create the (ring of quotients) extension \(Q\) of \(P\), and return the ideal \(J\) of \(Q\), together with the map \(f: P \rightarrow Q\).

If \(U\) has length \(k\) and the values (in order) of \(U\) are \(u_1, \ldots, u_k\), then first the rational function field \(F = K(x_{u_1}, \ldots, x_{u_k})\) is constructed, then the list \(v_1, \ldots, v_{n-k}\) is constructed as the list \(1, \ldots, n\) with the \(u_i\) removed, and finally the extension \(Q\) of \(P\) is defined to be the polynomial ring \(F[x_{v_1}, \ldots, x_{v_{n-k}}] = K(x_{u_1}, \ldots, x_{u_k})[x_{v_1}, \ldots, x_{v_{n-k}}]\).

The map \(f\) is constructed in the obvious way so that \(x_i\) is mapped to the appropriate variable in \(F\) if \(i\) is in \(U\), or the appropriate variable in \(Q\) otherwise. The image under \(f\) of an ideal of \(P\) is just the appropriate ideal of \(Q\) whose basis is obtained by taking the image under \(f\) of each of the polynomials in the basis of \(I\).

The inverse image under \(f\) of a polynomial of \(Q\) is obtained by first making the polynomial monic, then multiplying by the LCM of the denominators (“clearing the denominators”), then mapping each variable back to the appropriate one in \(P\)—this is possible since there are no proper denominators. The inverse image under \(f\) of an ideal \(H\) of \(Q\) is defined to be the ideal of \(P\) generated by the inverse images under \(f\) of the polynomials in the basis of \(H\) (note that this is not always equal to the contraction of \(H\)—see [Becker and Weispfenning, 1993, p. 389], for a simple algorithm to compute the contraction of an ideal of \(Q\)).