Finding small roots for bivariate polynomials over the ring of integers

In this paper, we propose the first heuristic algorithm for finding small roots for a bivariate equation modulo an ideal \begin{document}$ \mathcal{I} $\end{document} over the ring of integers \begin{document}$ \mathcal{R} $\end{document} . Existing algorithms for solving polynomial equations with size constraints only work for bivariate modular equations over integers, and univariate modular equation over number fields. Both previous algorithms use a relation between the short vector in a skillfully structured lattice and a size constrained solution. Our algorithm also follows this framework, but we additionally use a polynomial factoring algorithm over number fields to recover a 'ring' root of a bivariate polynomial equation. As a result, when an LLL algorithm is employed to find a short vector, we can recover all small roots of a bivariate polynomial modulo \begin{document}$ \mathcal{I} $\end{document} in polynomial time under some constraint.