Zeros and approximations of Holant polynomials on the complex plane
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We present fully polynomial approximation schemes for general classes of Holant problems with complex edge weights, which we call Holant polynomials. Our results are based on a recent technique for approximating graph polynomials of degree n by computing the first n coefficients of the Taylor expansion of the logarithm of the polynomial. Central to this technique is the absence of roots in regions on the complex plane. We establish these zero-free regions by translating our problem into partition functions of abstract geometric structures known as polymers in statistical physics. Our results are the first to consider Holant polynomials on the complex plane with such diverse classes of constraints. A noteworthy consequence of our research is an alternative angle on the problem of approximating the number of perfect matchings of general graphs - a central problem of unresolved complexity in the area of computational counting.
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