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quotient ring

asked 2017-10-02 15:54:37 -0500

malikb gravatar image

I need to compute some things in matrices over a quotient ring. My definition of the quotient ring in question is

v = 8

In particular, I need to quotient out by large (>8 shown above) powers of the ideal I. However, I've noticed that in constructing S, Sage much generate all of the monomials in the power of I which, when v=10, takes quite awhile. Is there any faster/more efficient way to have Sage internally generate S. It actually takes longer to construct S than it does to perform my computations.

This might just be how it is given I'm working over a polynomial ring with four indeterminates...

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answered 2017-10-02 21:02:17 -0500

dan_fulea gravatar image

One possibility to construct the ring would be as in the sample code:

sage: S = PowerSeriesRing( base_ring=QQ, names='a,b,c,d', default_prec=9 )
sage: S
Multivariate Power Series Ring in a, b, c, d over Rational Field
sage: a,b,c,d = S.gens()
sage: J = S.O(9)
sage: J
0 + O(a, b, c, d)^9
sage: u = (1+a^3+J)*(1+b^3+J)*(1+c^3+J)*(1+d^3+J)
sage: u
1 + a^3 + b^3 + c^3 + d^3 + a^3*b^3 + a^3*c^3 + a^3*d^3 + b^3*c^3 + b^3*d^3 + c^3*d^3 + O(a, b, c, d)^9
sage: 1/u
1 - a^3 - b^3 - c^3 - d^3 + a^6 + a^3*b^3 + a^3*c^3 + a^3*d^3 + b^6 + b^3*c^3 + b^3*d^3 + c^6 + c^3*d^3 + d^6 + O(a, b, c, d)^9

The above uses the PowerSeriesRing with a declared precision, and the ideal J is implemented as an $O$-calculus. Hope it works for the needed purpose.

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Thanks for this. This is quite promising, though I have one issue. Sage seems loathe to compute something as 0+O(...)^k, and so I'm getting monstrous terms that should simplify to 0 but aren't. Is there any way to have Sage fix the precision to be at most k-power terms so that 0 entries can be written? I can't seem to find any documentation about this.

malikb gravatar imagemalikb ( 2017-10-16 12:13:46 -0500 )edit

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Asked: 2017-10-02 15:54:37 -0500

Seen: 130 times

Last updated: Oct 02