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

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 QXnmQ\in X_n^m, let Q\mathbf Q be a form associated to QQ, meaning a highest-degree form of degree n(m2)n(m-2) on Cn\mathbb C^n arising from the nil-polynomial of the associated Artinian Gorenstein algebra.

Associated-form conjecture. For any IInm{\tt I}\in{\mathcal I}_n^m there exists an absolute invariant I\mathbf I of forms of degree n(m2)n(m-2) on Cn\mathbb C^n such that, for all QXnmQ\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.

Sources & referencesView supporting material

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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