TY - JOUR
T1 - On multivariate polynomials with many roots over a finite grid
AU - Geil, Olav
N1 - Publisher Copyright:
© 2021 World Scientific Publishing Company.
PY - 2021
Y1 - 2021
N2 - In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout's theorem, IEEE Trans. Inform. Theory 46(2) (2000) 635-641, On (or in) Dick Blahut's 'footprint', Codes, Curves Signals (1998) 3-9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.
AB - In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout's theorem, IEEE Trans. Inform. Theory 46(2) (2000) 635-641, On (or in) Dick Blahut's 'footprint', Codes, Curves Signals (1998) 3-9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.
KW - Finite field
KW - finite grid
KW - footprint bound
KW - multivariate polynomial
KW - root
KW - variety
UR - http://www.scopus.com/inward/record.url?scp=85094181680&partnerID=8YFLogxK
U2 - 10.1142/S021949882150136X
DO - 10.1142/S021949882150136X
M3 - Journal article
AN - SCOPUS:85094181680
SN - 0219-4988
VL - 20
JO - Journal of Algebra and its Applications
JF - Journal of Algebra and its Applications
IS - 8
M1 - 2150136
ER -