Expecting unexpected relations

The proposed project aims at exploring relations among powers of general linear forms with complex coefficients.

There is a long-standing conjecture due to Iarrobino from 1997 that states that there are no non-trivial relations among $d$-th powers of $m$ general linear forms, provided that the number of forms is at least $n+4$, where $n$ is the number of variables.The applicant has performed computer calculations which indicate that the conjecture is not true, and believes instead that relations will show up for any $n$ and any $m \geq n+4$ provided that $d$ is large enough.While the ultimate goal is to fully understand the relations that are expected to show up, there are several milestones on the way which would  have implications on contemporary mathematics.

The most direct application is in interpolation theory, and there is hope that the project will lead to generalizations of the Alexander-Hirschowitz Theorem, but also to proofs of special cases of the Segre-Gimigliano-Harbourne-Hirschowitz Conjecture.