Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

给定一个具有满足阿基米德性质版本的兼容前序的交换半环，由 Strassen (J. reine angew. Math. 1988) 引入的渐近谱是一个本质上独特的紧致 Hausdorff 空间以及从半环到连续函数环。Strassen 定理表征了前序的渐近松弛，该渐近比较了元素的大幂，直到作为相应函数的逐点偏序的次指数因子，将渐近谱实现为非负实数的单调半环同态空间。这种预先排序的半环已在复杂性理论和信息论中得到应用。我们证明了该定理对满足较弱多项式增长条件的预排序半环的推广。这个较弱的假设本身并不能确保非负实值单调同态表征（适当修改）渐近前序。我们找到了一个充分条件以及一个等价条件来支持它。在这些条件下，渐近谱是一个局部紧的 Hausdorff 空间，满足与 Strassen 工作中类似的普遍性质。