# System of Modular Bivariate Polynomials

I have two bivariate polynomials in two unknowns:

$x(y+a_{11}) + a_{12} \equiv b \pmod{m_1}$

$x(y+a_{21}) + a_{22} \equiv b \pmod{m_2}$

$x,y$ unknown

All $a_{ij}$ terms and the constant $b$ are relatively prime to both moduli, which may or may not be prime themselves, but have no common factors. I'm trying to solve these equations for all solutions - in some cases there are unique solutions, and in other cases not, but regardless I have not discovered an effective algorithm for finding the roots.

However I attempted to use a modified chinese remainder theorem that treated the right hand side as the term to solve for, however I ended up with third hyperbolic equation in $m'=m_1m_2$. This third equation has multiple solutions ($\phi(m')$ - both of the first two have $\phi(m_i)$ solutions individually as well), even in the case where the original system of equations would a unique solution.

I have also looked into multivariate CRTs such as this paper by Oliver Knill, and solutions to univariate and bivariate modular polynomials but these all rely on the moduli being the same for each equation; in this case that doesn't really apply. In the case of the CRT, I was able to get a similar equation in terms of $m'=m_1 m_2$, but this equation has multiple solutions, and only 1 seems to correspond to the system of equations.

I also attempted to take the inverse of x for each equation and came up with these related equations:

$a_{11}x^{-1}+ a_{12}y \equiv b_1 \pmod{m_1}$

$a_{21}x^{-1}+ a_{22}y \equiv b_2 \pmod{m_2}$

which is not all together helpful since if $x\pmod{m_1} = x \pmod{m_2}$ it is not necessarily the case that $x^{-1} \pmod{m_1} \ne x^{-1} \pmod{m_2}$

Any help would be appreciated.

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