The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop under field extensions, and both can jump in a limit. We show that, for fixed degree and under mild conditions on the characteristic of the ground field, the strength is at most a polynomial in the border strength. We also establish an analogous result for partition rank. Our results control both the jump under limits and the drop under field extensions.

翻译：多元齐次多项式的强度是指将其表示为低次齐次多项式乘积之和时所需项数的最小值。分区秩则是多线性形式的类似度量。这两种秩在域扩张下均可能下降，而在极限过程中均可能发生跃变。我们证明，在固定次数且基域特征满足温和条件的前提下，强度至多是边界强度的多项式函数。我们同样为分区秩建立了类似的结果。我们的研究结果同时控制了极限下的跃变与域扩张下的下降。