Bezout's Theorem and ideals of terminal forms

作者：Kang Ming chang^*

来源：Expositiones Mathematicae, 2010, 28(3): 265-268.

DOI：10.1016/j.exmath.2009.09.003

• Grobner basis
• Bezout's Theorem
• Macaulay's Theorem

Let k be any field, k[X(1),...,X(n)] be the polynomial ring of n variables over k. For any f = f(0) + f(1) + ... + f(r) is an element of k[X(1), ... ,X(n)] where each f(i) is a homogeneous polynomial of degree i and.f(r) not equal 0, define tm(f) = f(r). If I is an ideal in k[X(1),...,X(n)] define tm(I) to be <tm(f):f is an element of I\{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 is an element of k[X(1),X(2)] 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[X(1),...,X(n)] also.

• 出版日期2010