Suppose that R is a two-dimensional normal standard-graded domain over a finite field. We prove that there exists a uniform Frobenius test exponent b for the class of homogeneous ideals in R generated by at most n elements. This means that for every ideal I in this class we have that $f^{p^b}\in I^{[p^b]}$ if and only if $f\in I^{\rm F}$. This gives in particular a finite test for the Frobenius closure. On the other hand we show that there is no uniform bound for Frobenius test exponent for all homogeneous ideals independent of the number of generators. Under similar assumptions we prove also the existence of a bound for tight closure test ideal exponents for ideals generated by at most n elements.
