11.tex

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As an introduction to Gr\"obner basis, we first shall find the points lying on the surface defined by $S = g^{-1}(0)$ where $g(x,y,z) = x^4 + y^2 + z^2$; which are closest to $(1,1,1)$. In other words, this problem is about minimising the distance function (which we shall denote by $f$), and to do so, we shall make use of Lagrange multipliers method. This method states that this critical points $(x_0,y_0,z_0)$, can be found by means of the following equation
$$\nabla f(x_0,y_0,z_0) = \lambda\nabla g(x_0,y_0,z_0)$$

For our function, f(x,y,z) = $\sqrt{(x-1)^2 + (y-1)^2 + (z-1)^2}$, note that this function is non-degenerate and definite positive in $\R^3$ and therefore its critical points will be minimum points. Furthermore, the minimum points of $f$ will be the same as the ones for $f^2$. Since it is easier to work with $f^2$, we will henceforth denote by $f$, $f^2$. Then the system the equations that yields

$$\left\lbrace\begin{array}{l} 2(x-1) - 4\lambda x^3 = 0 \\ 
2(y-1) - 2\lambda y =0 \\ 2(z-1) - 2\lambda z = 0\\ x^4 + y^2 + z^2 -1 = 0\end{array}\right.$$
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