New conjectures on multiplicities of tensor eigenvalues

We work on two conjectures on tensor eigenvalue multiplicities. By using the language of algebraic geometry we give stronger, more refined versions of the conjectures and we prove them in many new cases, notably for all $2\times 2\times \cdots \times 2$ tensors. We also establish a connection between the rank of a tensor and the multiplicities of the zero eigenvalue.