How to make precise the notion of "the multiset of roots of a polynomial function"?

Question

A (real) polynomial function can be defined as a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a sequence $a : \mathbb{N} \rightarrow \mathbb{R}$ such that the terms of $a$ are ultimately zero, and for all $x \in \mathbb{R}$ it holds that $$f(x)=\sum_{i=0}^{\infty}a_ix^i.$$

We can also define a root-finding function $\rho$ that takes polynomial functions to multisets. For any polynomial function $f$, define that $\rho(f)$ is the multiset of all $x$ such that $f(x)=0$. However, this last statement is hideously imprecise. It makes sense to speak of, "The set of all $x$ such that ---condition---," but the multiset?

How might one go about defining it rigorously?

Answer 1

The root multiset arises naturally from the factorization of the polynomial

$$\rm f(x) = (x-r)^j \cdots (x-s)^k g(x)\ \to\ \{\, j\cdot r,\:\ldots,\: k\cdot s\,\}$$

where $\rm\:g(x)\:$ has no roots over the coefficient ring.

More precisely, define $\rm\ e_r(f(x)) := max\{n\in\Bbb N\ :\ (x\!−\!r)^n\!\mid f(x)\,\ in\,\ \Bbb R[x]\}.\:$ Then the root multiset is $\rm\ \{e_r(f)\cdot r\ :\ r\in roots(f)\},\:$ where $\rm\:n\cdot r\:$ denotes an element $\rm\:r\:$ of multiplicity $\rm\:n\:$ in a multiset.

Note that, over a domain, these linear factors are unique, being products of primes $\rm\,x-r.\:$ This uniqueness implies that the root multiset is well-defined. It also implies that any two (correct) root-finding algorithms will compute the same multiset of roots. The answer does not depend on what order the algorithm discovers the roots (as it generally does in nonunique factorization domains).

Answer 2

Not an answer but an alternative.

Instead of multiset, a rigorous way to deal with roots of polynomial over $\mathbb{C}$ with degree $n$ is to model the of roots of a polynomial as an element in a quotient space of $\mathbb{C}^n$. Two $n$-tuples $\lambda = (\lambda_1,\ldots,\lambda_n)$ and $\mu = (\mu_1,\ldots,\mu_n)$ are identified together if one can find a permutation $\sigma$ of the coordinates such that $\lambda_j = \mu_{\sigma(j)}$ for $j = 1,\ldots,n$.

The space is usually denoted as $\mathbb{C}_{sym}^n$. It inherits a natural quotient topology from $\mathbb{C}^n$ and has a natural metric calling optimal matching distance:

For $\lambda = (\lambda_1,\ldots,\lambda_n)$ and $\mu = (\mu_1,\ldots,\mu_n) \in \mathbb{C}_{sym}^n$, the optimal matching distance is defined as: $$d(\lambda,\mu) = \min_{\sigma} \max_{1\le j \le n} |\lambda_j - \mu_{\sigma(j)}|$$ where $\sigma$ runs over the set of permutations of the coordinates.

The most important point is under this metric/topology, the roots of a polynomial of degree $n$ depends continuously on its coefficients.