Linear transformation from polynomials

asked 2012-02-10 21:43:44 +0100

RPC
23 ●1 ●2 ●7

updated 2012-02-11 02:16:54 +0100

Suppose I have an unspecified list of degree 1 homogeneous polynomials in several variables, say [X1,X2,X3+3X4,X0]. This list will define a linear transformation [X0,X1,X2,X3,X4]|---->[X1,X2,X3+3X4,X0].

A priori I don't know how many variables or polynomials I will have, since they are found depending on some previous parameters. (The way I have done this, the variables are the generators of a polynomial ring V = PolynomialRing(QQ, dim,'X').)

My question is: How can I transform this list of polynomials into a matrix/linear transformation?

I've tried collecting the coefficients, but the .coefficients() does not work really well for multivariable polynomials since it does not "see the zero terms" (at least I don't know how to do that).

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Your question is not clear to me mathematically. What do you mean by transforming a list of polynomials into a matrix or linear transformation?

John Palmieri ( 2012-02-10 23:06:12 +0100 )edit

Sorry, I guess I was not too precise. I've edited and hopefully it will make sense now.

RPC ( 2012-02-11 02:17:30 +0100 )edit

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sage: dim=4 sage: F = PolynomialRing(QQ, dim,'X') sage: I = F.ideal([x*y for x,y in tuples(F.gens(),2)]) sage: pol = [I.reduce(F.random_element()) for i in range(dim)] sage: pol [-3*X2, -39/2*X1 - 1/18*X2, -1/10*X0 - 3*X1 - X2 - 1/5*X3, X2 + 6*X3] sage: matrix(dim,lambda i,j:pol[i].coefficient(F.gen(j))) [ 0 0 -3 0] [ 0 -39/2 -1/18 0] [-1/10 -3 -1 -1/5] [ 0 0 1 6]

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Thanks. That'll do it.

RPC ( 2012-02-11 16:45:53 +0100 )edit

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