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Simplification of expression with exponentials.

asked 2011-09-26 07:31:17 -0600

Eric Naslund gravatar image

Hello,

I am looking at expanding the product:

(a+b*g+c*g^2+d*g^3)*(a+b*g^2+c*g^4+d*g)*(a+b*g^3+c*g+d*g^4)*(a+b*g^4+c*g^3+d*g^2)

where $g=e^{2\pi i /5}$ is a fifth root of unity. The problem is, after expanding, and telling sage the value of $g$, it does not collect terms according to the rule $1+g+g^2+g^3+g^4=0$ so there are over 5 times too many terms.

Is there any way to make this simplification automatic? For degrees higher then $5$ going through manually will not be an option. (It is almost faster to multiply everything myself then figure out the collection of the terms)

Thank you very much for any help on this, I have been stuck for the last while

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answered 2011-09-26 11:36:53 -0600

Dirk Danckaert gravatar image

Coincidentally, something like this was the first thing I did with SAGE. Glad I can help you. It's really magic, but here's how it goes. Try to copy this in a SAGE notebook cell:

R=PolynomialRing(ZZ,'x,a,b,c,d')
x,a,b,c,d=R.gens()
S=R.quotient(1+x+x^2+x^3+x^4,'g,a,b,c,d')
g,a,b,c,d=S.gens()

Just to make sure, evaluate

g^5-1

(it should work out to 0). Then evaluate

(a+b*g+c*g^2+d*g^3)*(a+b*g^2+c*g^4+d*g)*(a+b*g^3+c*g+d*g^4)*(a+b*g^4+c*g^3+d*g^2)

and sure enough, the answer follows:

a^4 - a^3*b + a^2*b^2 - a*b^3 + b^4 - a^3*c + 2*a^2*b*c - 3*a*b^2*c -
b^3*c + a^2*c^2 + 2*a*b*c^2 + b^2*c^2 - a*c^3 - b*c^3 + c^4 - a^3*d +
2*a^2*b*d + 2*a*b^2*d - b^3*d - 3*a^2*c*d - a*b*c*d + 2*b^2*c*d +
2*a*c^2*d - 3*b*c^2*d - c^3*d + a^2*d^2 - 3*a*b*d^2 + b^2*d^2 +
2*a*c*d^2 + 2*b*c*d^2 + c^2*d^2 - a*d^3 - b*d^3 - c*d^3 + d^4
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Excellent Thank you!!! I had resorted to what I call a "half clever work around". I found $e^{2\pi i/5}$ to 16 decimal places, and expand as a polynomial with $g$ as a variable, and in the last step I substituted the numerical value for $g$. I then had to manually go through and remove the terms of the form 1.2123123 e^{-16} which was a major pain. This is so much better, thanks!

Eric Naslund gravatar imageEric Naslund ( 2011-09-26 11:46:32 -0600 )edit

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Asked: 2011-09-26 07:31:17 -0600

Seen: 160 times

Last updated: Sep 26 '11