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THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM

Part of: Commutative algebra: Homological methods Graph theory Extremal combinatorics Algebraic combinatorics

Published online by Cambridge University Press:  09 November 2020

NGUYEN CONG MINH Open the ORCID record for NGUYEN CONG MINH [Opens in a new window] ,
LUU BA THANG Open the ORCID record for LUU BA THANG [Opens in a new window]  and
TRAN NAM TRUNG Open the ORCID record for TRAN NAM TRUNG [Opens in a new window]
NGUYEN CONG MINH
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam e-mail: minhnc@hnue.edu.vn
LUU BA THANG*
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam
TRAN NAM TRUNG
Affiliation:
Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam and TIMAS, Thang Long University, Hanoi, Vietnam e-mail: tntrung@math.ac.vn
*
e-mail: thanglb@hnue.edu.vn
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Abstract

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Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

Keywords

combinatorial Nullstellensatzmultiplicitymultisetroot

MSC classification

Primary: 05D40: Probabilistic methods
Secondary: 05C90: Applications 05E40: Combinatorial aspects of commutative algebra 05E45: Combinatorial aspects of simplicial complexes 13D45: Local cohomology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 3 , June 2021 , pp. 369 - 378
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Footnotes

The third author is partially supported by NAFOSTED (Vietnam), grant number 101.04–2018.307.

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