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Declare arithmetic with formal variables

asked 2013-12-27 17:50:04 -0600

Zach H gravatar image

I want to create variables in SAGE and and declare arithmetic relations between them. For example, it is easy enough to declare that x1, ..., xn and y1, ..., yn be variables. Is there a way to state that

xi*xj = 0 (in the ring of polynomials, if necessary)

or that

x1 < y1 < x2 < y2 < ...?

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answered 2014-01-03 01:30:09 -0600

tmonteil gravatar image

In the algebraic way, you can do something like:

sage: R.<x,y> = PolynomialRing(ZZ) ; R
Multivariate Polynomial Ring in x, y over Integer Ring
sage: I = R.ideal(x*y)
sage: I
Ideal (x*y) of Multivariate Polynomial Ring in x, y over Integer Ring
sage: S = R.quotient_ring(I) ; S
Quotient of Multivariate Polynomial Ring in x, y over Integer Ring by the ideal (x*y)
sage: S(3*x*y+x^2+4*y)
xbar^2 + 4*ybar

In the symbolic way, you can do (but it is much less reliable):

sage: var('x y')
(x, y)
sage: assume(x*y==0)
sage: bool(3*x*y+x^2+4*y == x^2+4*y)
True

But is seems not able to decide simplifications by itself:

sage: (3*x*y+x^2+4*y).full_simplify()
x^2 + (3*x + 4)*y

For the orderings, you can also work symbolically:

sage: var('x y')
(x, y)
sage: assume(x<y)
sage: bool(3*x<3*y)
True
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Asked: 2013-12-27 17:50:04 -0600

Seen: 148 times

Last updated: Jan 03 '14