The Nash-Tognoli theorem asserts that every compact smooth manifold $M$ of dimension $d$ is smoothly diffeomorphic to a nonsingular algebraic subset $X$ of $\mathbb{R}^{2d+1}$, a so-called algebraic model of $M$. Since the subset $X$ of $\mathbb{R}^n$ is algebraic and nonsingular, it can be described both globally and locally by a finite number of polynomials with real coefficients, that is, there exist polynomials $p_1,\ldots,p_n$ in $\mathbb{R}[x_1,\ldots,x_{2d+1}]$ for a certain positive natural number $n$ such that:

• $X$ is the locus of the zeros of the polynomials $p_i$, that is, $X=\{x\in\mathbb{R}^{2d+1}:p_1(x)=0,\ldots,p_n(x)=0\}$ and
• locally at every point $y$ of $X$, the polynomials $p_i$ describe the differential structure of $X$ via the Implicit Function Theorem, that is, for every $y\in X$, there exist a neighborhood $U$ of $y$ in $\mathbb{R}^{2d+1}$ and $d+1$ indices $i_1,\ldots,i_{d+1}$ extracted from $\{1,\ldots,n\}$ (that depend on $y$) such that $X\cap U=\{x\in U:p_{i_1}(x)=0,\ldots,p_{i_{d+1}}(x)=0\}$ and the gradients of the polynomials $p_{i_1},\ldots,p_{i_{d+1}}$ evaluated at $y$ are linearly independent.