In the classical linear degeneracy testing problem, we are given $n$ real numbers and a $k$-variate linear polynomial $F$, for some constant $k$, and have to determine whether there exist $k$ numbers $a_1,\ldots,a_k$ from the set such that $F(a_1,\ldots,a_k) = 0$. We consider a generalization of this problem in which $F$ is an arbitrary constant-degree polynomial, we are given $k$ sets of $n$ numbers, and have to determine whether there exist a $k$-tuple of numbers, one in each set, on which $F$ vanishes. We give the first improvement over the na\"ive $O^*(n^{k-1})$ algorithm for this problem (where the $O^*(\cdot)$ notation omits subpolynomial factors). We show that the problem can be solved in time $O^*\left( n^{k - 2 + \frac 4{k+2}}\right)$ for even $k$ and in time $O^*\left( n^{k - 2 + \frac{4k-8}{k^2-5}}\right)$ for odd $k$ in the real RAM model of computation. We also prove that for $k=4$, the problem can be solved in time $O^*(n^{2.625})$ in the algebraic decision tree model, and for $k=5$ it can be solved in time $O^*(n^{3.56})$ in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for $k$-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft's point-line incidence detection problem in any dimension.
翻译:在古典线性变异测试问题中,我们得到的是美元真实数字和美元变异的线性多元数字,对于一些恒定美元美元来说,我们需要确定是否存在美元数字($a_1,\ldots,a_k$),从这个设置中确定是否存在美元数字($a_1,\ldots,a_k)=0美元。我们考虑对这一问题的概括化,其中F美元是任意的常态多调多式,我们得到的是美元数($),我们得到的是美元数($),我们得到的是美元($)的直线线线线线线线线线线线线线线线($),我们得到的是美元数组数组数组数字($美元),每套一套数(美元),我们得到的是美元(美元)的元(美元) 元(美元) 元(美元) 元(美元) 元(美元(美元) 平价(美元) 平价(美元) 平价(美元(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元) 平价(美元)