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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 $R^n$.

Edit. I just realized that Sergei Ivanov gave a very elementary proof in his answer herehere.

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 $R^n$.

Edit. I just realized that Sergei Ivanov gave a very elementary proof in his answer here.

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 $R^n$.

Edit. I just realized that Sergei Ivanov gave a very elementary proof in his answer here.

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Misha
Misha
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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 $R^n$.

Edit. I just realized that Sergei Ivanov gave a very elementary proof in his answer here.

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 $R^n$.

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 $R^n$.

Edit. I just realized that Sergei Ivanov gave a very elementary proof in his answer here.

Source Link
Misha
Misha
  • 31.2k
  • 1
  • 94
  • 163

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 $R^n$.