The associated-form conjecture for invariants of degree-\n(m-2) forms

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Let XnmX_n^m be the space of degree-mm forms in nn variables, and let Inm{\mathcal I}_n^m denote the relevant algebra of invariants of such forms. For Q∈XnmQ\in X_n^m, let Q\mathbf Q be a form associated to QQ, meaning a highest-degree form of degree n(m−2)n(m-2) on Cn\mathbb C^n arising from the nil-polynomial of the associated Artinian Gorenstein algebra.

Associated-form conjecture. For any I∈Inm{\tt I}\in{\mathcal I}_n^m there exists an absolute invariant I\mathbf I of forms of degree n(m−2)n(m-2) on Cn\mathbb C^n such that, for all Q∈XnmQ\in X_n^m, the invariant I\mathbf I is defined at some (hence at every) form Q\mathbf Q associated to QQ and

I(Q)=I(Q).\mathbf I(\mathbf Q)={\tt I}(Q).

The conjecture proposes that every invariant of degree-mm forms can be recovered from an absolute invariant of their associated forms. It was essentially verified for binary quartics (n=2n=2, m=4m=4) and ternary cubics (n=3n=3, m=3m=3), but the general statement is resolved according to the supplied status evidence.

References

Primary source

M. G. Eastwood and A. V. Isaev, “Invariants of Artinian Gorenstein Algebras and Isolated Hypersurface Singularities”, arXiv:1402.6049 (2014).

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