We present a new algorithm for solving homogeneous multilinear equations, which are high dimensional generalisations of solving homogeneous linear equations. First, we present a linear time reduction from solving generic homogeneous multilinear equations to computing hyperdeterminants, via a high dimensional Cramer's rule. Hyperdeterminants are generalisations of determinants, associated with tensors of formats generalising square matrices. Second, we devise arithmetic circuits to compute hyperdeterminants of boundary format tensors. Boundary format tensors are those that generalise square matrices in the strictest sense. Consequently, we obtain arithmetic circuits for solving generic homogeneous boundary format multilinear equations. The complexity as a function of the input dimension varies across boundary format families, ranging from quasi-polynomial to sub exponential. Curiously, the quasi-polynomial complexity arises for families of increasing dimension, including the family of multipartite quantum systems made of $d$ qubits and one qudit.

Update with extended version that was accepted to ITCS 2025. A few errors in the previous version are corrected and some directions towards proving (cryptographic) hardness of hyperdeterminants are added.

We present a new algorithm for solving homogeneous multilinear equations, which are high dimensional generalisations of solving homogeneous linear equations. First, we present a linear time reduction from solving generic homogeneous multilinear equations to computing hyperdeterminants, via a high dimensional Cramer's rule. Hyperdeterminants are generalisations of determinants, associated with tensors of formats generalising square matrices. Second, we devise arithmetic circuits to compute hyperdeterminants of boundary format tensors. Boundary format tensors are those that generalise square matrices in the strictest sense. Consequently, we obtain arithmetic circuits for solving generic homogeneous boundary format multilinear equations. The complexity as a function of the input dimension varies across boundary format families, ranging from quasi-polynomial to sub exponential. Curiously, the quasi-polynomial complexity arises for families of increasing dimension, including the family of multipartite quantum systems made of $d$ qubits and one qudit.