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Characterization of the Hilbert-Samuel polynomials of curve singularities. (English) Zbl 0724.13021
Let A be a one-dimensional Cohen-Macaulay local ring, (b,e,$$\rho$$) a triplet of integers. Let $$\rho_{0,b,e}=(r+1)e-\left( \begin{matrix} r+b\\ r\end{matrix} \right)$$, where r is the integer such that $$\left( \begin{matrix} b+r-1\\ r\end{matrix} \right)\leq e<\left( \begin{matrix} b+r\\ r+1\end{matrix} \right)$$ and $$\rho_{1,b,e}=e(e-1)/2-(b-1)(b-2)/2$$. The main result of the paper is that there exists a one-dimensional Cohen-Macaulay local ring A with embedding dimension b, multiplicity e and reduction number $$\rho$$ iff $$b=e=1$$ and $$\rho =0$$ or $$2\leq b\leq e$$ and $$\rho_{0,b,e}\leq \rho \leq \rho_{1,h,e}$$. In this last case, one can choose A to be the quotient of a power series ring over an algebraically closed field of characteristic zero. Using this, the Hilbert-Samuel function of one- dimensional Cohen-Macaulay rings of small multiplicity is computed. Lastly, conditions for the Cohen-Macaulayness of gr(A) are established in terms of the reduction number.

##### MSC:
 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14H20 Singularities of curves, local rings 13C14 Cohen-Macaulay modules
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