See
S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scu. Norm. di Pisa, 18 (1964), 449-474.
where it is proven that real algebraic (actually, even semi-analytic) sets admit stratifications whose strata are smooth manifolds. Of course, you need to assume that your polynomial is nonconstant, to ensure that there are no strata open in $R^n$.