Numerical Root Finding via Cox Rings

We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification $X$ of $(\mathbb{C} \setminus \{0\})^n$. We homogenize the input equations to obtain a homogeneous ideal in the Cox ring of $X$ and generalize the eigenvalue, eigenvector theorem for rootfinding in affine space to compute homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method. In particular, the method outperforms existing solvers in the case of (nearly) degenerate systems with solutions on or near the torus invariant prime divisors.

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