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Extended Euclid with polynomials

asked 2014-01-03 12:31:55 -0500

erinb gravatar image

updated 2014-01-03 14:19:49 -0500

calc314 gravatar image

Suppose given polynomials $e,q,h,r$ in $R[x]$, $p \in R$ (R a ring), how can I use Sage to find $f$ in $R[x]$ so $f e = q h + r (\text{mod } p)$?

Similarly, given $f,g$ in $R[x]$ with $\text{gcd}(f,g)=1$, what function can I use to compute $s,t$ in $R[x]$ so $s f + g h = 1 (\text{mod } p)$ ?

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answered 2014-01-03 14:27:56 -0500

wdoo gravatar image

p=7 R=PolynomialRing(IntegerModRing(7),'x') x=R.gen()

I'm not sure if f exist

Maybe f=(q*h+r)//e

Or you want something like this? q=f//g r=f.mod(g)

As for the next question:

f=x^14 + 4*x^7 + 7

g=x^3+1

xgcd(f,g)

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answered 2014-01-09 11:12:05 -0500

erinb gravatar image

updated 2014-01-09 11:12:26 -0500

Thanks!

and is there a way to work around the use of the integermodring? I'm writing an algorithm in which I have to succesively compute different functions e.g.

s_j g_j + t_j h_j = 1 modulo p^2^j
s_j g_j + t_j h_j = 1 modulo p^2^(j+1)

Would it be best to redefine

R=PolynomialRing(IntegerModRing(p^2^j),'x') every time?

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Asked: 2014-01-03 12:31:55 -0500

Seen: 462 times

Last updated: Jan 09 '14