In the very first chapter Hartshorne proposes the following seemingly trivial exercise (ex. I.2.17(ii)):

Show that a strict complete intersection is a set theoretic complete intersection.

Here are Hartshorne's definitions:

A variety $Y$ of dimension $r$ in $\mathbb{P}^n$ is a (strict) complete intersection if $I(Y)$ can be generated by $n-r$ elements. $Y$ is a set-theoretic complete intersection if $Y$ can be written as the intersection of $n-r$ hypersurfaces.

Here $I(Y)$ is the homogeneous ideal of $Y$. The point is that the first definition seems wrong, since one would naturally require that $I(Y)$ can be generated by $n-r$ *homogeneous* elements (with this definition the exercise becomes trivial).

I have never made my mind if this is a misprint by Hartshorne. So the question is

Is it true that in a polynomial ring any homogeneous ideal generated by $k$ elements is also generated by $k$ homogeneous elements?

If I recall well it is not difficult to find counterexamples in graded rings which are not polynomial rings, so the point of the exercise may be to show that polynomial rings have this special property.