Counting solutions to binomial complete intersections

E Cattani, University of Massachusetts - Amherst
A Dickenstein

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This is the pre-published version harvested from ArXiv. The published version is located at http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHX-4K7FJCX-1&_user=1516330&_coverDate=02%2F28%2F2007&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1662372887&_rerunOrigin=google&_acct=C000053443&_version=1&_urlVersion=0&_userid=1516330&md5=fe0f34aeb86639c32ede298f34158305&searchtype=a

Abstract

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.