A polynomial function in one variable is a function $f$ such that $$f(x) = \sum_n a_nx^n.$$ This definition is purely formal in that it gives a form for a polynomial function, but no characteristics. In contrast, definitions for other things (such as linear function, differentiable function) don't give a particular form or template, but rather a set of properties.
Is there a similar definition for a polynomial function? A polynomial function is a function that...
One candidate might be: is analytic, defined everywhere, and has a finite number of zeroes. However, this would not exclude $e^x$ which has zero zeroes, just like $f(x) = k$.