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Find a root of bivariate polynomial

asked 2023-04-08 07:07:42 +0100

toran gravatar image

Given a bivariate and symmetric polynomial $P(x,y)$ with a high degree (probably larger than 8). Is there any algorithm that helps me know if $P(x,y)$ has a root over $\mathbb{R}^+$ or not? I may not need a specific root, I just want to check if there is a root, or not.

*P/S: I'm new to SageMath. Currently, I know some basic operation with multivariate polynomial like factorizing, adding, substracting, ... two polynomials. It would be great if you can explain your answer a little bit. Thank you.

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answered 2023-04-12 02:26:50 +0100

Max Alekseyev gravatar image

updated 2023-04-12 02:52:30 +0100

Potentially this can be solved with QEPCAD package (which needs to be installed first). Here is an example:

x,y = var('x y')
P = x^3 + y^2 - 1

qf = qepcad_formula.and_( P == 0, x >= 0, y >= 0 )
    root = qepcad(qf,solution='any-point')
except ValueError:
    print('P has no root in R+')
    print(f'P has a root: {root}')

It can be run at Sagecell.

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Asked: 2023-04-08 07:06:42 +0100

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Last updated: Apr 12 '23