Abstract: We prove that a singular real-analytic Levi-flat hypersurface $H$ in ${\mathbb{C}}^n$ is
Segre-degenerate at a point $p$ is equivalent to the existence of a so-called
support curve, that is, a holomorphic curve that intersects $H$ at exactly
one point, which is equivalent to the existence of support curves on at least
two sides of $H$ at $p$. The existence of such two-sided support provides families
of analytic discs attached to $H$ which covers a neighborhood of $p$. The existence
of such discs has two corollaries. First,
any function holomorphic on a neighborhood of a Segre-degenerate $H$ extends to a fixed
neighborhood of $p$. Second, the polynomial hull of $H$ is a neighborhood of $p$,
and thus no Levi-flat Segre-degenerate hypersurface in ${\mathbb{C}}^n$ can be polynomially
convex.
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