On the complication of algebras of combinant invariants (Q1764844)

Let \(\mathbb{S}_{d}\) be the \(\text{SL}_{2}( \mathbb{C})\)-module of degree \(d\) binary forms. Then \(\text{SL}_{2}( \mathbb{C} )\) acts on the cone of the Grassmannian of subspaces of a fixed dimension \(k\) in \(\mathbb{S}_{d},\) i.e. SL\(_{2}( \mathbb{C})\) acts on \(A_{k,d},\) the cone of \(\mathbb{G}_{k}( \mathbb{S}_{d}).\) This paper examines the complication of the invariants of \(A_{k,d}^{\text{SL}_{2}( \mathbb{C})},\) the invariants combinants. This complication is denoted cpl \(A_{k,d}^{\text{SL}_{2}( \mathbb{C} )}.\) There are two results presented here. The first is that, for any natural number \(n\) there are only finitely many pairs of integers \(k\) and \(d\) such that cpl \(A_{k,d}^{\text{SL}_{2}( \mathbb{C})}<n.\) The second result gives the explicit conditions for when cpl \(A_{k,d}^{\text{SL} _{2}(\mathbb{C}) }<15\).