Mathematics – Commutative Algebra
Scientific paper
2010-01-04
Electron. J. Combin. 17(1) (2010), Note 14
Mathematics
Commutative Algebra
4 pages
Scientific paper
It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba (O. Kouba, A duality based proof of the Combinatorial Nullstellensatz. Electron. J. Combin. 16 (2009), #N9) one obtains a direct proof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz for polynomials over an arbitrary integral domain. The proof relies on Cramer's rule and Vandermonde's determinant to explicitly describe a map used by Kouba in terms of cofactors of a certain matrix. That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon's work and emphasized in recent articles of Michalek and Schauz; the sole purpose of the present note is to point out that not only is it not necessary to invoke duality of vector spaces, but by not doing so one easily obtains a more general result.
No associations
LandOfFree
Proof of the combinatorial nullstellensatz over integral domains in the spirit of Kouba does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Proof of the combinatorial nullstellensatz over integral domains in the spirit of Kouba, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Proof of the combinatorial nullstellensatz over integral domains in the spirit of Kouba will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-7550