Exercise 10.8

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

(continuation) If $f$ is homogeneous, a point $\overline{a}$ on the hypersurface defined by $f$ is said singular if it is simultaneously a zero of all the partial derivatives of $f$. If the degree of $f$ is prime to the characteristic, show that a common zero of all the partial derivatives of $f$ is automatically a zero of $f$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
If $\frac{\partial f}{\partial x_i} (\overline{a}) = 0$ for all $i = 1,\ldots,n$, then $mf(\overline{a}) = \sum\limits_{i=1}^n x_i \frac{\partial f}{\partial x_i} (\overline{a}) = 0$. Since $m = \deg(f)$ is prime with the characteristic, then $m$ is non zero in the field $F$, thus $f(\overline{a}) = 0$.
\end{proof}

Exercise 10.8

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Exercise 10.8

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