Mathematical Research Letters

Volume 20 (2013)

Number 4

The $j$-multiplicity of monomial ideals

Pages: 729 – 744



Jack Jeffries (Department of Mathematics, University of Utah, East Salt Lake City, Ut., U.S.A.)

Jonathan Montaño (Department of Mathematics, Purdue University, Lafayette, Indiana, U.S.A.)


We prove a characterization of the $j$-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissier’s volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals. We also give a description of the $\epsilon$-multiplicity of a monomial ideal in terms of the volume of a region.

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