In this paper, we focus on computing the kernel of a map of polynomial rings $\varphi$. This core problem in symbolic computation is known as implicitization. While there are extremely effective Gr\"obner basis methods used to solve this problem, these methods can become infeasible as the number of variables increases. In the case when the map $\varphi$ is multigraded, we consider an alternative approach. We demonstrate how to quickly compute a matrix of maximal rank for which $\varphi$ has a positive multigrading. Then in each graded component we compute the minimal generators of the kernel in that multidegree with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where Gr\"obner techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is \emph{embarassingly parallel} and a fully parallelized version of our algorithm will be forthcoming in OSCAR.

翻译：本文聚焦于计算多项式环映射 $\varphi$ 的核。这一符号计算的核心问题被称为隐式化。虽然存在极为有效的Gröbner基方法来解决此问题，但随着变量数量的增加，这些方法可能变得不可行。针对映射 $\varphi$ 为多梯度的情况，我们考虑了一种替代方法。我们展示了如何快速计算一个最大秩矩阵，使得 $\varphi$ 具有正多梯度性。随后在每个梯度分量中，我们通过线性代数计算出该多度数下降核的最小生成元。我们已在Macaulay2中实现了所提出的技术，并表明在Gröbner技术失效的实例中，我们的实现能够计算出大量低度数的生成元。这包括多个来自系统发生学的例子，其中甚至二次和三次生成元的完整列表此前尚未可知。当多梯度细化总度数时，我们的算法具有天然的并行性，其完全并行化版本将在OSCAR中推出。