# The Cayley-Salmon theorem via classical differential geometry

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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