About characteristics of graded algebras \(S_{1,4}\) and \(SI_{1,4}\). (English) Zbl 1204.34043

Gherstega, N.; Mihail, Popa; Victor, Pricop

Bul. Acad. Ştiinţe Repub. Mold., Mat. 2010, No. 1(62), 23-32 (2010).

Summary: We consider a differential system of the form
\[ \dot x^j=a^j_\alpha x^\alpha+a^j_{\alpha\beta\gamma\delta}x^\alpha x^\beta x^\gamma x^\delta\quad (j,\alpha,\beta,\gamma,\delta=1,2), \tag{1} \]
where the coefficient tensor \(a^j_{\alpha\beta\gamma\delta}\) is symmetrical in lower indices in which the complete convolution holds.
One method to study the set of centro-affine invariants and comitants of system (1) is the method of generating functions and Hilbert series.
It is known that in order to construct a minimal polynomial base of invariants and comitants of system (1), it is enough to construct generators of algebra of unimodular comitants and invariants.
We construct Hilbert series for the graded algebras of comitants \(S_{1,4}\) and invariants \(SI_{1,4}\) of the differential system (1) and we determine with their help the Krull dimensions of these algebras. Lower bounds for the number of the types of generators for the algebras \(S_{1,4}\) and \(SI_{1,4}\) are obtained.

MSC:

34C14

Symmetries, invariants of ordinary differential equations

Keywords:

differential system; invariants and comitants of differential systems; Hilbert series; dimension of Krull