# What are the sizes of 'functions' and 'fields' in the context of input to a decision problem?

I can reduce 3-Sat to the following NP-Complete decision problem:

Let $$S =$${0,1,...,s-1}, $$D \subseteq S$$ and $$P$$ be a multivariate polynomial in $$n$$ variables. Decide yes iff there exists $$\vec x \in D^{n}$$ such that $$P(\vec x) \neq 0 (mod \ s)$$.

That got me thinking that I can make the problem seem much harder by rephrasing the problem as:

Let $$\mathbb{F}$$ be a field, $$f$$ be some function over $$\mathbb{F}$$ and $$D$$ be a restriction of the domain of $$\mathbb{F}$$. Decide yes iff there exists $$x \in D$$ such that $$f(x) \neq 0$$.

With this new problem, I don't know the size of the inputs. If I restrict myself to polynomials and modular arithmetic, the functions can only admit operations done in polynomial time, so their "sizes" seem obvious. How can I talk about the time complexity of deciding or verifying that a function is not identically 0? e.g. $$sin(1) \neq 0$$ in $$\mathbb{R}$$

Or, to put it in a more pedantic way, time complexity is associated with formal languages. A formal language $$L$$ is defined as a subset of $$\{0,1\}^*$$. So, to speak precisely, you first need to specify a well-defined language $$L$$; once you have done that, you can talk about its time complexity, but if you're not clear on what the language is, then it does not have a well-defined time complexity.
But you have to be careful. Some kinds of mathematical objects don't necessarily have any concise representation, so the details of how they are represented make a big difference. In your example, it is not clear how to represent $$f$$. Or, one natural way to represent a function is as a truth table -- but this representation can be very large, so the resulting time complexity (as a function of the length of the input) might not capture what you really want to know about.