Past work shows that one can associate a notion of Shannon entropy to a Dirichlet polynomial, regarded as an empirical distribution. Indeed, entropy can be extracted from any d:Dir by a two-step process, where the first step is a rig homomorphism out of Dir, the *set* of Dirichlet polynomials, with rig structure given by standard addition and multiplication. In this short note, we show that this rig homomorphism can be upgraded to a rig *functor*, when we replace the set of Dirichlet polynomials by the *category* of ordinary (Cartesian) polynomials. In the Cartesian case, the process has three steps. The first step is a rig functor PolyCart -> Poly sending a polynomial p to (dp)y, where dp is the derivative of p. The second is a rig functor Poly -> Set x Set^op, sending a polynomial q to the pair (q(1),Gamma(q)), where Gamma(q)=Poly(q,y) can be interpreted as the global sections of q viewed as a bundle, and q(1) as its base. To make this precise we define what appears to be a new distributive monoidal structure on Set x Set^op, which can be understood geometrically in terms of rectangles. The last step, as for Dirichlet polynomials, is simply to extract the entropy as a real number from a pair of sets (A,B); it is given by log A - log B^(1/A) and can be thought of as the log aspect ratio of the rectangle.

翻译：先前研究表明，可以将香农熵的概念与狄利克雷多项式（视为经验分布）相关联。事实上，熵可通过两步过程从任意d:Dir中提取：第一步是构建从Dir（狄利克雷多项式集合，其rig结构由标准加法与乘法定义）出发的rig同态。本文中，我们证明当将狄利克雷多项式集合替换为普通（笛卡尔）多项式范畴时，该rig同态可升级为rig函子。在笛卡尔情形下，该过程包含三个步骤：第一步是PolyCart→Poly的rig函子，将多项式p映射为(dp)y（其中dp为p的导数）；第二步是Poly→Set×Set^op的rig函子，将多项式q映射为配对(q(1),Γ(q))，其中Γ(q)=Poly(q,y)可解释为视作纤维丛的q的整体截面，而q(1)为其基空间。为精确表述该过程，我们定义了Set×Set^op上一种全新的分配幺半结构，该结构可从矩形几何角度理解。最后一步（与狄利克雷多项式情形相同）从集合对(A,B)中提取熵为实数，其表达式为log A - log B^(1/A)，可视为矩形的长宽比对数。