I would like to express "there exists a function that the number of terms of some polynomial that describes the function is smaller than some defined number" in set-theoretic/logic way (first-order sentence). Can this be done?

Also, can we say in first-order sentence that "there exists a function that every term is smaller than some defined number"?

(Clarification: By the number of terms, I mean: $f(x) = x^3+2x+2$ has three terms - $x^3, 2x, 2$.)