The Cayley-Salmon theorem via classical differential geometry

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Let $latex {f: {bf R}^3 rightarrow {bf R}}&fg=000000$ be an irreducible polynomial in three variables. As $latex {{bf R}}&fg=000000$ is not algebraically closed, the zero set $latex {Z_{bf R}(f) = { x in{bf R}^3: f(x)=0}}&fg=000000$ can split into various components of dimension between $latex {0}&fg=000000$ and $latex {2}&fg=000000$. For instance, if $latex {f(x_1,x_2,x_3) = x_1^2+x_2^2}&fg=000000$, the zero set $latex {Z_{bf R}(f)}&fg=000000$ is a line; more interestingly, if $latex {f(x_1,x_2,x_3) = x_3^2 + x_2^2 – x_2^3}&fg=000000$, then $latex {Z_{bf R}(f)}&fg=000000$ is the union of a line and a surface (or the product of an acnodal cubic curve with a line). We will assume that the $latex {2}&fg=000000$-dimensional component $latex {Z_{{bf R},2}(f)}&fg=000000$ is non-empty, thus defining a real surface in $latex {{bf R}^3}&fg=000000$. In particular, this hypothesis implies that $latex {f}&fg=000000$ is not just irreducible over $latex {{bf R}}&fg=000000$, but is in fact absolutely irreducible (i.e. irreducible over $latex {{bf C}}&fg=000000$), since…

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