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Suppose that $0$ is a regular value of $f$. Then $0$ is a critical value of $g=f^2$, yet the level set $g^{-1}(0)$ is a regular submanifold. In this case all the points on $g^{-1}(0)$ are critical points of $g$.

The general answer is difficult. You need to assume something about $f$ f$. A natural assumption would be that the critical points of$f$are isolated and of finite type. Near such points one can find local coordinates so that in these coordinate$f$looks like a polynomial. (This is a generalization of the classical Morse lemma due to Tougeron.) In such cases you need to understand the zero sets of real polynomials which can be challenging. A nice place toconsult to consult for such issues is the book by Arnold, Gussein-Zade and Varchenko on singularities of differentiable mappings. 1 Suppose that$0$is a regular value of$f$. Then$0$is a critical value of$g=f^2$, yet the level set$g^{-1}(0)$is a regular submanifold. The general answer is difficult. You need to assume something about$f$A natural assumption would be that the critical points of$f$are isolated and of finite type. Near such points one can find local coordinates so that in these coordinate$f\$ looks like a polynomial. (This is a generalization of the classical Morse lemma due to Tougeron.) In such cases you need to understand the zero sets of real polynomials which can be challenging. A nice place toconsult for such issues is the book by Arnold, Gussein-Zade and Varchenko on singularities of differentiable mappings.