Symmetric spatial curves
The geometry of symmetric spatial curves reveals characterisations of general one-parameter families of complex univariate polynomials with fully-symmetric Galois groups.
Bernstein-Kouchnirenko-Khovanskii with a symmetry
A generic polynomial f(x,y,z) with a prescribed Newton polytope defines a symmetric spatial curve f(x,y,z)=f(y,x,z)=0. We study its geometry: the number, degree and genus of its irreducible components, the number and type of singularities, etc. and discuss to what extent these results generalize to higher dimension and more complicated symmetries. As an application, we characterize generic one-parameter families of complex univariate polynomials, whose Galois group is a complete symmetric group.