Quadratic polynomial/R/Variable set/Pure form/Normed standard form/Remark

A quadratic form in standard form

$\sum _{1\leq i\leq k}r_{i}U_{i}^{2}+s\,\,{\text{ or }}\,\,\sum _{1\leq i\leq k}r_{i}U_{i}^{2}+sU_{k+1}$

in the sense of fact can be simplified further, if we allow distortions. In the new coordinates

${}Z_{i}={\sqrt {\vert {r_{i}}\vert }}U_{i}\,$

or

${}U_{i}={\frac {1}{\sqrt {\vert {r_{i}}\vert }}}Z_{i}\,$

for ${}i=1,\ldots ,k$, the quadratic form has a representation of the form

$\sum _{1\leq i\leq k}\pm Z_{i}^{2}+s\,\,{\text{ or }}\,\,\sum _{1\leq i\leq k}\pm Z_{i}^{2}+sZ_{k+1},$

the coefficients are ${}1$ or ${}-1$. This is called the normed standard form of the quadratic form. By swapping of the variables, we may achieve that the first variables have the coefficient ${}1$, and the later variables have coefficient ${}-1$. In doing these transformations, the vanishing sets are distorted. For example, an ellipse might become a circle, or a parabola might be compressed. Since the vanishing set does not change by multiplying the form with ${}-1$, we may also assume that the number variables with coefficient ${}1$ is at least the number of variables with coefficient ${}-1$.