Hilbert polynomials for the algebra of invariants of binary d-form

Nadia Ilash


We consider one of the fundamental problems of classical invariant theory - the research  of Hilbert polynomials for the algebra of invariants  of the Lie group SL2. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. It is well-known that the Hilbert function of the algebra SLn-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary d-form Cd=Vd2SL2 (here Vd is the  d+1-dimensional space of binary forms of degree d) was obtained by Sylvester. Then it was  generalized to  algebra of  invariants of  binary d-form Id=VdSL2. But the Cayley-Sylvester formula is  not  expressed in terms of polynomials. In our article we consider the problem of computing  the Hilbert polynomials for the algebra  of invariants of  binary d-form.

Повний текст:

PDF (English)


L. Bedratyuk, The Poincare series of the covariants of binary forms, Int. J. Algebra 4 (2010), no. 25-28, 1201-1207.

L. Bedratyuk, The Poincare series of the algebras of simultaneous invariants and covariants of two binary forms, Linear Multilinear Algebra. 58 (2010), no. 6, 789-803. DOI: 10.1080/03081080903127262

L. Bedratyuk, Weitzenbock derivations and the classical invariant theory, I: Poincare series, Serdica Math. J. 36 (2010), no. 2, 99-120.

L. Bedratyuk, Analogue of the Cayley-Sylvester formula and the Poincare series for the algebra of invariants of n-ary form, Linear Multilinear Algebra 59 (2011), no. 11, 1189-1199. DOI: 10.1080/03081081003621303

L. Bedratyuk, The Poincare series for the algebras of invariants of binary and ternary forms, Naukovi zapysky NaUKMA. Fiz.-Mat. Nauky 113 (2011), 7-11 (in Ukranian).

L. Bedratyuk, Hilbert polynomials of the algebras of SL2-invariants, arXiv:1102.3290v1, 2011, Preprint.

P. de Carvalho Cayres Pinto, H.-C. Herbig, D. Herden, and C. Seaton, The Hilbert series of SL2-invariants, Commun. Contemp. Math. 22 (2020), no. 7, Art. ID 1950017, pp. 38. DOI: 10.1142/S0219199719500172

D. Eisenbud, The geometry of syzygies. A second course in commutative algebra and algebraic geometry, Springer, New York, 2005. DOI: 10.1007/b137572

D. Hilbert, Theory of algebraic invariants, Cambridge University Press, 1994.

N. B. Ilash, Hilbert polynomials of the algebras of SL1-invariants, Carpathian Math. Publ. 10 (2018), no. 2, 303-312. DOI: 10.15330/cmp.10.2.303-312

L. Robbiano, Introduction to the theory of Hilbert function, Curves Semin. Queen’s. Vol. VII, Queen’s Pap. Pure Appl. Math. 85, Expose B, (1990), 26 p.

T. A. Springer, On the invariant theory of SU2, Indag. Math. 42 (1980), no. 3, 339-345. DOI: 10.1016/1385-7258(80)90034-7

R. P. Stanley, Hilbert functions of graded algebras, Adv. Math. 28 (1978), no. 1, 57-83. DOI: 10.1016/0001-8708(78)90045-2

DOI: http://dx.doi.org/10.30970/vmm.2021.92.077-085


  • Поки немає зовнішніх посилань.