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Foundations of Computational Mathematics

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30-06-2016

About Gordan’s Algorithm for Binary Forms

Author: Marc Olive

Published in: Foundations of Computational Mathematics | Issue 6/2017

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Abstract

In this article, we present a modern viewpoint on the Gordan algorithm for binary forms. The symbolic method is recast in terms of \(\mathrm {SL}(2,{\mathbb {C}})\) equivariant homomorphisms. A graphical approach is used to define Gordan’s ideal, a central tool used to obtain an integrity basis for the covariant algebra of a binary form. To illustrate the power of this method, we compute for the first time a minimal integrity basis for the covariant algebra of \(\mathrm {S}_{6}\oplus \mathrm {S}_{4}\), \(\mathrm {S}_{6}\oplus \mathrm {S}_{4}\oplus \mathrm {S}_{2}\) and for the invariant algebra of \(\mathrm {S}_{8}\oplus \mathrm {S}_{4}\oplus \mathrm {S}_{4}\).

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There exist several methods to compute this Hilbert series [10, 48, 61] a priori.

Note that Weyman [71] has also reformulated Gordan’s method in a modern way and through algebraic geometry, but unfortunately, we were unable to extract from it an effective approach. There is also a preprint of Pasechnik [58] on this method.

For a general and modern approach on invariant and covariant algebras, we refer to the online text [44] by Kraft and Procesi.

A general overview of what is known about invariants/covariants of binary forms is available at http://www.win.tue.nl/~aeb/math/invar.html.

This operator is called scaling process in [54].

It is important to note that a digraph represents here a morphism and not a bi-differential operator as in Olver–Shakiban [55].

The covariant \({\mathbf {M}}^{\nu (r)}\) is called a term in [38].

There is an explicit expression for these coefficients in [51].

This operation is called convolution in [38].

For simplicity, we can suppose that \(\kappa \) is the sum of two irreducible solutions.

Equivalently (by 4.3 and 4.7), we can choose the set of all molecular covariants with atoms taken in \(\mathrm {A}\).

Such an ideal is clearly an homogeneous ideal as being generated by homogeneous elements.

This means that for \(k'<k\), we have \(({\mathbb {C}}[{\mathcal {F}}_k])_{k'}={\mathcal {A}}_{k'}\), and if we take a strict subfamily \({\mathcal {G}}\varsubsetneq {\mathcal {F}}_k\), this property is no more true.

Such a covariant basis had been used to obtain a covariant basis of \(\mathbf {Cov}(\mathrm {S}_{10})\) [53].

We note here \(\mathbf {Inv}_{j}(V_1\oplus V_2)\) a set of joint invariants of degree \(d_1>0\) and \(d_2>0\) in \(V_1\) and \(V_2\).

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Title: About Gordan’s Algorithm for Binary Forms
Author: Marc Olive
Publication date: 30-06-2016
Publisher: Springer US
Published in: Foundations of Computational Mathematics / Issue 6/2017
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-016-9324-x

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