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<p>A well-known theorem by Milnor-Orlik provides a formula for the Milnor number of a weighted-homogeneous polynomial having an isolated singularity that depends only on the weights. In this paper we present a proof of that result using techniques from commutative algebra. Our approach is to obtain a free resolution of the Milnor algebra through the Koszul complex. The desired formula is then obtained from a Hilbert series calculation.</p>

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An algebraic proof of the Milnor-Orlik theorem

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  • 出版模式：

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  • 发表日期：

    2026年5月20日

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  • Yerly Soler

摘要

A well-known theorem by Milnor-Orlik provides a formula for the Milnor number of a weighted-homogeneous polynomial having an isolated singularity that depends only on the weights. In this paper we present a proof of that result using techniques from commutative algebra. Our approach is to obtain a free resolution of the Milnor algebra through the Koszul complex. The desired formula is then obtained from a Hilbert series calculation.