Visualizing properties of comprehensive SAGBI bases. -- Two examples (Q5949978)

The author has studied SAGBI (subalgebra analogue of Gröbner bases for ideals) bases of rings of invariants of conjugates of permutation groups acting on polynomial rings [\textit{M. Göbel}, J. Symb. Comput. 19, No. 4, 285-291 (1995; Zbl 0832.13006); \textit{M. Göbel} and \textit{J. Walter}, J. Algorithms 32, No. 1, 58-61 (1999; Zbl 0946.13004)]. In the latter paper, the authors conjectured that if \(G\) is a subgroup of the symmetric group \(S_n\), then there exists a matrix \(\delta \in \text{GL}(n,K)\) such that \(K[X_1,\dots,X_n]^{G^\delta}\) has a finite SAGBI basis with respect to the lexicographical order, where \(G^\delta\) denotes the conjugate of \(G\) by \(\delta\). In the paper under review, the author considers the special case of \({\mathbb C}[X_1,X_2,X_3]^{G^L}\), where \(G\) is \(A_3\) or \(S_3\) and \(L=\left(\begin{smallmatrix} 1&0&0\\ a&1&0\\\;b&c&1\end{smallmatrix}\right)\). He presents plots of the locus of points \((a,b,c)\) such that the above ring of invariants has a finite SAGBI basis. In the case of \(G=S_3\), the experimental evidence is that all finite reduced SAGBI bases have either 3 or 4 elements and that the locus of points \((a,b,c)\) such that the reduced SAGBI basis contains 4 elements consists of the union of four planes. All calculations were done with a MATHEMATICA package that is available from the author.