Counting Real Roots in Polynomial-Time for Systems Supported on Circuits
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Suppose $A={a_1,ldots,a_{n+2}}subsetmathbb{Z}^n$ has cardinality $n+2$, with all the coordinates of the $a_j$ having absolute value at most $d$, and the $a_j$ do not all lie in the same affine hyperplane. Suppose $F=(f_1,ldots,f_n)$ is an $n imes n$ polynomial system with generic integer coefficients at most $H$ in absolute value, and $A$ the union of the sets of exponent vectors of the $f_i$. We give the first algorithm that, for any fixed $n$, counts exactly the number of real roots of $F$ in in time polynomial in $log(dH)$.