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Filtering an expression: keeping only term with even power

asked 2012-10-22 12:26:25 -0500

Nicolas Essis-Breton gravatar image

updated 2012-10-22 13:27:36 -0500

In the expression $$ x^2 y^2+x y^2+x^3y + 5 x^4 y^4 $$ I would like to keep only the term where the variables have even power $$ x^2y^2+5 x^4 y^4 $$

Is there a way to do it?

I look into this post.
But once I get the operands, I don't see how to analyze the variable exponent.

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answered 2012-10-22 13:30:05 -0500

achrzesz gravatar image

updated 2012-10-22 20:12:59 -0500

Is that what you need?

var('x y');
p=(5*x^2*y^2+x*y^2+x^3*y+x^4*y^4).polynomial(QQ)          
sum([v[0]*v[1] for v in p if forall(v[1].exponents()[0],lambda x:is_even(x))[0]])
#x^4*y^4 + 5*x^2*y^2
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@archrzesz Exactly what I want, thank you. Is there a way to keep the coefficient of each monomial? For example, return $5 x^4 y^4$.

Nicolas Essis-Breton gravatar imageNicolas Essis-Breton ( 2012-10-22 13:42:54 -0500 )edit

I've edited my answer according to the Nicolas request. Notice that my answer works for many variables. Just change var and p.

achrzesz gravatar imageachrzesz ( 2012-10-22 20:14:50 -0500 )edit

@achrzesz Very flexible approach. Thank you.

Nicolas Essis-Breton gravatar imageNicolas Essis-Breton ( 2012-10-23 00:35:57 -0500 )edit
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answered 2012-10-22 21:27:21 -0500

calc314 gravatar image

The following also works, although it might not be as flexible as the answer by @achrzesz, which handles things nicely.

x, y = var('x, y')
f(x,y)=x^2*y^2-4*x^2*y+2*y^2+4*x*y^4-3*x^4*y^6
g(x,y)=(f(x,y)+f(-x,y))/2
h(x,y)=(g(x,y)+g(x,-y))/2
h
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answered 2012-10-22 13:46:14 -0500

calc314 gravatar image

updated 2012-10-22 13:47:32 -0500

Here is an answer using the list command and some list comprehensions.

var('x y')
R.<x,y>=PolynomialRing(RR)
q=x^2*y^2+x+x^2*y+y^2
terms=list(q)
ans=[t[0]*t[1] for t in terms if (t[1].degree(x) % 2)==0 and (t[1].degree(y) % 2==0)]
sum(ans)

This will also keep the coefficients.

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Asked: 2012-10-22 12:26:25 -0500

Seen: 174 times

Last updated: Oct 22 '12