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Depth One Homogeneous Prime Ideals in Polynomial Rings over a Field
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This paper concerns the question: Which depth one homogeneous prime ideals N in a polynomial ring H are of the principal class? In answer to this question, we introduce acceptable bases of ideals in polynomial rings, and then use a known one-to-one correspondence between the ideals N in H := F[X1, . . . , Xn] such that Xn ∉ N and the maximal ideals P in the related polynomial ring G := F[X1/Xn, . . . , Xn−1/Xn] to show that the acceptable bases of the maximal ideals P in G transform to homogeneous bases. This is used to determine several necessary and sufficient conditions for a given depth one homogeneous prime ideal N in H to be an ideal of the principal class, thus answering, in part, our main question. Then it is shown that the Groebner-grevlex bases of ideals are acceptable bases. Finally, we construct several examples to illustrate our results, and we delve deeper into an example first studied by Macaulay.
Keywords
Ideal Basis, Polynomial Ring, Prime Ideal.
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