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Extension and Contraction of Ideals

Extension and Contraction of Ideals

Magma allows the extension to and contraction from the ring of quotients of an ideal with respect to certain variables. See Becker & Weispfenning, pages 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 = R[x_1, ..., x_n], 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 -> Q. If U has length k and the values (in order) of U are u_1, ..., u_k, then first the rational function field F = R(x_(u_1), ..., x_(u_k)) is constructed, then the list v_1, ..., v_(n - k) is constructed as the list 1, ..., n with the u_i removed, and finally the extension Q of P is defined to be the polynomial ring F[x_(v_1), ..., x_(v_(n - k))] = R(x_(u_1), ..., x_(u_k))[x_(v_1), ..., 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 & Weispfenning, page 389, for a simple algorithm to compute the contraction of an ideal of Q).
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