Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find easy counterexamples to classical statements. I hope somebody may help me to clarify. On page 69, Proposition 3.3.1.4 it states "Let $V$ be a variety of dimension $d$. Then the subset of its singular points is a Zariski subset of dimension strictly less than $d$." By variety it is meant the common zero locus of finitely many polynomials in $n$ variables with coefficients in $\mathbb{R}$. By dimension it is meant the Krull dimension of the quotient by the radical ideal generated by them of $\mathbb{R}[X_1, ..., X_n]$. This looks more or less the same as in the classical framework on algebraically closed fields of coefficients. Now, if we take $V=Z(X^2 + Y^2)$ over $\mathbb{R}$, I understand that the polynomial $X^2 + Y^2$ in $\mathbb{R}[X,Y]$ is irreducible over $\mathbb{R}$. Also, $V$ has dimension $1$ accordingly with the definition above ($(X^2 + Y^2)\subset (X,Y)$ maximal with no prime ideals in between). Now, it is clear that $V=\{(0,0)\}=Z(X,Y)$, with the last one the zero locus of $\nabla (X^2 + Y^2)$, hence all points of $V$ seem to be singular? So how is it possible that the singular points form a subset of strictly lower dimension?
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