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We describe conjecturally the generalized Samuel multiplicitiesof a monomial idealin terms of its Newton polyhedronMore precisely, we conjecture that  c iequals the sum of the normalized-volumes of pyramids over the projections of the-dimensional compact faces ofalong the infinite-directions of  i -unbounded facets in which they are contained. For  c0  proofs are known (Guibert, Jeffries and Montaño) and fora proof is given.

Generalized Samuel Multiplicities of Monomial Ideals and Volumes