Suppose $A=\{a_1,\ldots,a_{n+2}\}\subset\mathbb{Z}^n$ 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 $n\times n$ 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 an algorithm that, for any fixed $n$, counts exactly the number of roots of $F$ in the positive orthant in time polynomial in $\log(dH)$. (The fastest previous algorithms had exponential dependence on $\log d$, already for $n=2$.) The use of Diophantine approximation over number fields, to identify the underlying discriminant chamber containing $F$, plays a key role. Our underlying estimates are also useful for certifying numerical solutions of certain sparse polynomial systems.

中文翻译：

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电路支持的系统在多项式时间内计算正根

假设$ A = \ {a_1，\ ldots，a_ {n + 2} \} \ subset \ mathbb {Z} ^ n $具有基数$ n + 2 $，而$ a_j $的所有坐标的绝对值为大多数$ d $和$ a_j $都不都位于同一仿射超平面中。假设$ F =（f_1，\ ldots，f_n）$是一个$ n \ n的多项式系统，其泛型整数系数的绝对值最大为$ H $，而$ A $为该函数的指数向量集的并集$ f_i $。我们给出一种算法，对于任何固定的$ n $，它会精确计算$ \ log（dH）$中时间正多项式中$ F $的根数。（先前最快的算法已经对$ \ log d $进行了指数依赖，对于$ n = 2 $而言。）使用Diophantine逼近数字字段来识别包含$ F $的基础判别腔，起着关键作用。