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zcA;I`)5!jLE=c~V`M9_EU2w-wel(4Jv<6tph1kbcYdrRmyEE{t7XSPliz_}F+TDI~eBd6*sckI5`tFGwTM(k0X-R@hg~`Py#D~GaAg``~@&5pxL$&_^ diff --git a/ws2022/rav/lecture/rav.tex b/ws2022/rav/lecture/rav.tex index e6fbd7c..0a50d12 100644 --- a/ws2022/rav/lecture/rav.tex +++ b/ws2022/rav/lecture/rav.tex @@ -37,5 +37,6 @@ Christian Merten (\href{mailto:cmerten@mathi.uni-heidelberg.de}{cmerten@mathi.un \input{rav21.tex} \input{rav19.tex} \input{rav20.tex} +\input{rav22.tex} \end{document} diff --git a/ws2022/rav/lecture/rav22.pdf b/ws2022/rav/lecture/rav22.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bcbe255fb9ababa65a46fde880022f4ce477956d GIT binary patch literal 115025 zcmbTeby(Ev8a6sK4Ba6e0#Y*!-Q5iWBHbWe0@4j4-6bs`sephe-Cc?T(%mT_FlSuL zwXgH-HRo*o;JVO1f6x8Y{nX2(AtT2D=Hvx3jZgQ?0tIP7w9e-CKoJojmx`5>jfX8Q zL=Xa|{p$zFC1>a0VdX~4CFfw~VI^Z_;cRII6cYowd$?JdIRd@E?i)UGzt@K!@VQ1J z4_OzZs2R68aNA*<=J~ZhqFc>33+FEkF$4Zfr9==m*E5q@;Mv69=>4iSOq$Hxku)_G`V^4-(p$K 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Then the ideal + $\mathcal{I}(Z)$ is a real ideal. +\end{satz} + +\begin{proof} + If $Z = \emptyset$, then $\mathcal{I}(Z) = \mathcal{I}(\emptyset) = k[T_1, \ldots, T_n]$ is + a real ideal. Now assume $Z \neq \emptyset$. In this case, if $P_1, \ldots, P_r \in k[T_1, \ldots, T_n]$ + and $\lambda_1, \ldots, \lambda_r > 0$ in $k$ are such that + $\sum_{j=1}^{r} \lambda_j P_j^2 \in \mathcal{I}(Z)$, then + for all $x \in Z$, $\sum_{j=1}^{r} \lambda_j P_j^2(x) = 0$ in $k$. Since + $k$ is an ordered field and $\lambda_j > 0$ for all $j$, this implies + that for all $j$, $P_j(x) = 0$, i.e. $P_j \in \mathcal{I}(Z)$. +\end{proof} + +Recall that if $k$ is an arbitrary field and $I \subsetneq k[T_1, \ldots, T_n]$ is a proper ideal, +then finding a common zero $x \in L^{n}$ to all polynomials $P \in I$ for some extension $L$ of $k$ +is equivalent to finding a homomorphism of $k$-algebras +\[ +\varphi\colon k[T_1, \ldots, T_n]/I \longrightarrow L +.\] Indeed, the correspondence is obtained by sending such a $\varphi$ +to $x = (x_1, \ldots, x_n)$ where $x_i = \varphi(T_i \text{ mod } I)$. The basic +result should be about giving sufficient conditions for such homomorphisms to exist. + +\begin{theorem}[Real Nullstellensatz I] + Let $(k, \le )$ be an ordered field and let $k^{(r)}$ be the real closure of $k$. Let + $I \subseteq k[T_1, \ldots, T_n]$ be a real ideal. Then there exists a homomorphism + of $k$-algebras + \[ + k[T_1, \ldots, T_n] / I \longrightarrow k^{(r)} + .\] In particular, if $I \subsetneq k[T_1, \ldots, T_n]$ is a proper real ideal, then + $\mathcal{V}_{k^{r}}(I) \neq \emptyset$. + \label{thm:real-nullstellensatz} +\end{theorem} + +Let $(k, \le)$ be an ordered field. For the proof of \ref{thm:real-nullstellensatz}, we need +two lemmata: + +\begin{lemma} + Let $I \subseteq k[T_1, \ldots, T_n]$ be a real ideal. Then $\sqrt{I} = I$. Moreover, + if $\mathfrak{p} \supset I$ is a minimal prime ideal containing $I$, then + $\mathfrak{p}$ is real. +\end{lemma} + +\begin{lemma} + Let $\mathfrak{p} \subseteq k[T_1, \ldots, T_n]$ be a prime ideal. Then the fraction field + \[ + K \coloneqq \operatorname{Frac}\left( k[T_1, \ldots, T_n]/\mathfrak{p} \right) + \] is a real field if and only if the prime ideal $\mathfrak{p}$ is real. In that case + $K$ can be ordered in a way that extends the order of $k$. +\end{lemma} + +\end{document}