使用少量算术运算评估几何查询
Evaluating geometric queries using few arithmetic operations
摘要：
设$\cp:=(P_1,...,P_s)$是给定的整系数$n$ -变量多项式族，并假设这些多项式的次数和对数高度分别由$d$和$h$有界.此外，假设对于每个$1\leq i\leq s$，多项式$P_i$可以使用$L$算术运算（加法、减法、乘法以及常数0和1）来评估。假设族$\cp$在适当的意义上是\emph{generic}。我们构建了一个数据库$\cal D$，由代数计算树支持，这样对于每个$x\in [0,1]^n$查询$P_1(x),...,P_s(x)$的符号可以回答使用$h d^{\cO(n^2)}$比较和$nL$实数之间的算术运算。的算术几何工具开发的建设$\cal D$，然后展示的例子类的系统的$n$多项式方程的$n$未知数的一致性可以检查使用只有很少的算术运算，但承认指数数量的比较。
英文摘要：
Let $\cp:=(P_1,...,P_s)$ be a given family of $n$-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by $d$ and $h$, respectively. Suppose furthermore that for each $1\leq i\leq s$ the polynomial $P_i$ can be evaluated using $L$ arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family $\cp$ is in a suitable sense \emph{generic}. We construct a database $\cal D$, supported by an algebraic computation tree, such that for each $x\in [0,1]^n$ the query for the signs of $P_1(x),...,P_s(x)$ can be answered using $h d^{\cO(n^2)}$ comparisons and $nL$ arithmetic operations between real numbers. The arithmetic-geometric tools developed for the construction of $\cal D$ are then employed to exhibit example classes of systems of $n$ polynomial equations in $n$ unknowns whose consistency may be checked using only few arithmetic operations, admitting however an exponential number of comparisons.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
