Abstract
We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci.
Highlights
Let f ∈ Rd = K[x, y]d be a binary form of degree d, where K = R or C
The case of binary forms was considered and completely solved by Sylvester [36], who proved that the generic rank, i.e., the complex rank of a general complex binary form of degree d, is d +1 2
The generic complex rank of forms in more variables is described by the celebrated Alexander–Hirschowitz theorem [1]
Summary
A rank is called typical for real binary forms of degree d if it occurs in an open subset of Rd , with respect to the Euclidean topology. It is irreducible when d is odd, and it has two irreducible components when d is even From these general results, it follows a complete description of all the algebraic boundaries with low degree d ≤ 6. We show in [5] that all the boundaries between two typical ranks are unions of dual varieties to suitable coincident root loci. Coincident root loci are well-studied varieties which parametrize binary forms with multiple roots, see Sect. We study the algebraic boundaries for forms of arbitrary degree, and our main result is the following: Theorem 0.1.
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