17 citations · 36 across the 10 of their papers we have counts for
10 papers
Bounding the signed count of real bitangents to plane quartics
Mario Kummer, Stephen McKean
Using methods from enriched enumerative geometry, Larson and Vogt gave a signed count of the number of real bitangents to real smooth plane quartics. This signed count depends on a…
Circles of Apollonius two ways
Stephen McKean
Because the problem of Apollonius is generally considered over the reals, it suffers from variance of number: there are at most eight circles simultaneously tangent to a given trio…
Lifts, transfers, and degrees of univariate maps
Thomas Brazelton, Stephen McKean
One can compute the local -degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that f…
Conics meeting eight lines over perfect fields
Cameron Darwin, Aygul Galimova, Miao Pam Gu +1
Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an e…
Bézoutians and the -degree
Thomas Brazelton, Stephen McKean, Sabrina Pauli
We prove that both the local and global -degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that…
Rational lines on smooth cubic surfaces
Stephen McKean
We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth…
Bézoutians and injectivity of polynomial maps
Stephen McKean
We prove that an endomorphism of affine space is injective on rational points if its Bézoutian is constant. Similarly, is injective at a given rational point if its reduced…
An arithmetic enrichment of Bézout's Theorem
Stephen McKean
The classical version of Bézout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using wo…
All lines on a smooth cubic surface in terms of three skew lines
Stephen McKean, Daniel Minahan, Tianyi Zhang
Jordan showed that the incidence variety of a smooth cubic surface containing 27 lines has solvable Galois group over the incidence variety of a smooth cubic surface containing 3 s…
The trace of the local -degree
Thomas Brazelton, Robert Burklund, Stephen McKean +2
We prove that the local -degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local -deg…