Bezout's Theorem and ideals of terminal forms
Resource
Expositiones Mathematicae, 28(3),265-268
Journal
Expositiones Mathematicae
Journal Volume
28
Journal Issue
3
Pages
265-268
Date Issued
2010
Author(s)
Abstract
Let k be any field, k [X1, ..., Xn] be the polynomial ring of n variables over k. For any f = f0 + f1 + ⋯ + fr ∈ k [X1, ..., Xn] where each fi is a homogeneous polynomial of degree i and fr ≠ 0, define tm (f) = fr. If I is an ideal in k [X1, ..., Xn], define tm (I) to be 〈 tm (f) : f ∈ I {minus 45 degree rule} { 0 } 〉, the ideal generated by the terminal forms tm (f). Using Bezout's Theorem and Macaulay's Theorem, we will establish the following. If f, g ∈ k [X1, X2] satisfying that gcd { f, g } = gcd { tm (f), tm (g) } = 1 and I = 〈 f, g 〉, then tm (I) = 〈 tm (f), tm (g) 〉. Actually the above result is equivalent to Bezout's Theorem, which sheds another perspective of Bezout's Theorem. These results are valid in k [X1, ..., Xn] also. © 2009 Elsevier GmbH. All rights reserved.
Type
journal article
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