# Is there any method to solve a Bivariate Cubic Equation System? [closed]

$f(x, y) = 0$ and $g(x, y) = 0$, both $f$ and $g$ are cubic polynomial equation (at most 10 coefficients for each).

Is there any fixed method to solve this degenerate equation system? thanks.

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WHat's degenerate about it? And what do you mean by "solve"? Numerically? In radicals? –  Igor Rivin Jul 30 '13 at 13:43

## closed as unclear what you're asking by Andrey Rekalo, David White, Daniel Moskovich, John Pardon, Chris GodsilJul 30 '13 at 18:15

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Resultants are good, but for practical computations the Bezout Lemma (it's the same as the extended Euclidean algorithm) can be less expensive: Assume that $f(x,y)$ and $g(x,y)$ are relatively prime. Consider $f$ and $g$ as polynomials in $y$ over the field $k(x)$, where $k$ is your (unspecified) base field. Then there are $r,s\in k(x)[y]$ with $r(y)f(x,y)+s(y)g(x,y)=1$. Multiplying with the least common multiple $u(x)$ of the denominators of the coefficients of $r(y)$ and $s(y)$ yields $R(x,y)f(x,y)+S(x,y)g(x,y)=u(x)$, where $R,S\in k[x,y]$. So if $f=g=0$, then $u=0$.

Of course $u(x)$ is essentially the resultant of $f$ and $g$ with respect to $y$.

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