In dice notation, any collection of polyhedral dice can be represented as a polynomial in one unknown. For example, $1d^{20} + 3d^6$ denotes a collection consisting of one 20-sided die and three 6-sided dice.
There is a map $P$ sending any such collection of dice to the corresponding probability distribution. For example, the map sends the single $k$-sided die $d^k$ to the uniform distribution $U_k$ over $1\ldots k$.
In general, you compute the distribution of a sum of dice by taking the convolution of the individual dice distributions. For example, the map $P$ sends $d^a + 3d^b$ to $U_a\oplus (U_b \oplus U_b \oplus U_b)$.
In this way, the map $P$ is a morphism between the space of dice polynomials (formalized as a polynomial rig) and the space of dice probabilities (formalized as the set of all the uniform distributions $U_1, U_2, U_3,\ldots$, plus all finite convolutions of them).
It preserves addition, sending polynomial sums to probability convolutions.
But polynomials also have multiplicative structure: The product of two individual dice $d^a\cdot d^b$ is $d^{a+b}$, for example, and the corresponding distribution is $U_{a+b}$.
My question: Can we define multiplication on the target space of distributions so that $P$ preserves multiplication as well as addition, acting as a kind of ring homomorphism?
We already know what the multiplication operator must be on the uniform distributions $U_k$; we have $U_a \otimes U_b \equiv U_{a+b}$.
I think that if we impose the distributive property along with this base case, we have enough to define $\otimes$ throughout the entire space. Indeed if $A$ and $B$ are finite convolutions of $U_i$, then $A\otimes B$ can be expanded according to the distributive property into a unique convolution of uniform distributions $U_j$.
But I am unsure how to prove whether this works---is the operator really well defined, or does it somehow give contradictory or underdetermined answers in some cases?
2026-03-27 17:52:36.1774633956