Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz
We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the geometric degree of the system of equations. The obtained bound is polynomial in these pa...
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1997
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00224049_v117-118_n_p565_Sombra http://hdl.handle.net/20.500.12110/paper_00224049_v117-118_n_p565_Sombra |
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paper:paper_00224049_v117-118_n_p565_Sombra2023-06-08T14:50:33Z Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the geometric degree of the system of equations. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases. The proof of this result is combinatorial, and relies on global estimates for the Hilbert function of homogeneous polynomial ideals. In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical polynomial ideal. © 1997 Elsevier Science B.V. 1997 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00224049_v117-118_n_p565_Sombra http://hdl.handle.net/20.500.12110/paper_00224049_v117-118_n_p565_Sombra |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| description |
We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the geometric degree of the system of equations. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases. The proof of this result is combinatorial, and relies on global estimates for the Hilbert function of homogeneous polynomial ideals. In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical polynomial ideal. © 1997 Elsevier Science B.V. |
| title |
Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz |
| spellingShingle |
Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz |
| title_short |
Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz |
| title_full |
Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz |
| title_fullStr |
Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz |
| title_full_unstemmed |
Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz |
| title_sort |
bounds for the hilbert function of polynomial ideals and for the degrees in the nullstellensatz |
| publishDate |
1997 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00224049_v117-118_n_p565_Sombra http://hdl.handle.net/20.500.12110/paper_00224049_v117-118_n_p565_Sombra |
| _version_ |
1768546334247223296 |