In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group G, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are G-equivalent. We prove that for any G-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group and some quantities depending on a choice of classifying invariants. We show that all generic curves have signatures of the same degree and this degree is the sharp upper bound. For the full projective group, as well as for its affine, special affine and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve.2 Differential invariants and signatures of algebraic curvesThe set of all rational G-invariant functions is denoted by C(Y) G . It is easy to see that it is a subfield of the field C(Y) of all rational functions on Y.The set W is called a domain of separation for I.Due to the Noetherian property, there exists a maximal (with respect to inclusions) domain of separation. It is not difficult to see that a maximal domain of separation is a union of orbits, and therefore is a G-invariant set.In the following proposition, we summarize several important and non-trivial results about the structure of C(Y) G . See [36] or [42] for details.Definition 2.8. We say that an algebraic curve X ⊂ C 2 is G-equivalent to an algebraic curve Y ⊂ C 2 if there exists g ∈ G such that X = g · Y .Clearly G-equivalence satisfies all properties of an equivalence relation, and we use the notation X ∼ = G Y to denote the G-equivalence of curves X and Y . Elements g ∈ G defining self-equivalences of X are called symmetries of X in G. It is not difficult to show that the set of all symmetries Sym(X, G) = {g ∈ G | X = g · X} form a closed algebraic subgroup of G, called the symmetry group of X with respect to G.2 In [42], this result is attributed to Rosenlicht. 3 i.e. isomorphic to a field of rational functions of a finite number of independent variables. 4 In [42], this result is attributed to Lüroth and Castelnuovo. 5 From now on we will refer to PGL(3) as the projective group.
In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of n × n Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is 2n − 3.
We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to noise.
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