Counting solutions to binomial complete intersections

  • Cattani E
  • Dickenstein A
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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. © 2006 Elsevier Inc. All rights reserved.

Author-supplied keywords

  • # P-complete
  • Binomial ideal
  • Complete intersection

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  • Eduardo Cattani

  • Alicia Dickenstein

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