Field degrees and multiplicities for non-integral extensions (Q2382947)

Let \(S=k[t_1, \cdots, t_d]\) be a standard graded polynomial ring over a field \(k\) and consider the \(k\)--subalgebra \(R=k[x_1, \cdots, x_{m}]\) generated by the homogeneous elements \(\{x_1, \cdots, x_{m}\}\). Let \(| x| \) denote the degree of a homogeneous element \(x\). Suppose \(S\) is algebraic over \(R\). The problem of interest in this paper is to calculate the degree \([S:R]\) of the underlying field extension in terms of \(\{| x_1| , \cdots, | x_m| \}\). A well known theorem of Bézout states that if \(S\) is integral over \(R\) then (a) \([S:R]\) divides \(\prod | x_i| \) and (b) if \(m=d\) then \([S:R]=\prod | x_i| \). The authors deal with the converse of part (b). They prove that if \(S\) is algebraic over \(R\) then (a) \([S:R] \leq \prod | x_i| \) and (b) if \([S:R] \geq \prod | x_i| \) then \(S\) is integral over \(R\) (Theorem~1.2). Note that if \(S\) is not integral over \(R\) then \([S:R]\) may not divide \( \prod | x_i| \) even if \(m=d\) as can be seen in Example~5.1. The proof of Theorem~1.2 is reduced to another result that obtains a criterion for integrality in terms of multiplicities (Theorem~1.3). Theorem~1.3 is of independent interest and can be also obtained as a special case of a result of \textit{A. Simis, B. Ulrich}, and \textit{W. Vasconcelos} [Math. Proc. Cambridge Philos. Soc 130, No. 2, 237--257 (2001; Zbl 1096.13500)] (Theorem~6.1). Here the assumptions are stronger and the proof is more detailed. In Section 5 they give an application of Theorem~1.2 to rings of invariants (Theorem~1.4) and in Section~6 they provide a series of examples of rings of invariants, where Theorem~1.4 provides a proof of polynomial structure.