I am looking for a way to solve 3rd and 4th degree polynomials from Python (actually Python 3) using Sage. The polynomials are not square-free and the modulo can be a composite number. How do I do that? Help would be appreciated.
Like this:
mymod = 21
x = polygen(Zmod(mymod))
f = x^4 + x^2 + 1
print( f.roots(multiplicities=False) )
Hey, awesome. I can't believe you answered so quickly. Thanks. One more question. What if I know the factorization, e.g. 7*3, can I pass it in to speed up the calculation? I guess it can be just precomputed once.
I believe it will be factored once if you create a single Zmod (and x) object and reuse it in all polynomials.
You can get the multiplicity of root r as valuation(f, x-r). Note however that modulo a composite number polynomial roots may exhibit quite an unusual behavior - e.g., the total multiplicity of all roots may be larger than the polynomial degree. Even in my example x^4+x^2+1modulo 21 has 12 distinct roots. See also https://math.stackexchange.com/q/2823339
I got TypeError: Polynomial_dense_modn_ntl_ZZ.valuation() takes no arguments (1 given) for my example. It works fine with yours.
Asked: 2025-03-21 12:10:10 +0200
Seen: 100 times
Last updated: Mar 21
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