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Let $f$ be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given $f$ for this action is known to be a finite group. We determine explicit families that parameterize all endomorphisms defined over $bar{mathbb{Q}}$ of degree $3$ and degree $4$ that have a nontrivial automorphism, the textit{automorphism locus} of the moduli space of dynamical systems. We analyze the geometry of these loci in the appropriate moduli space of dynamical systems. Further, for each family of maps, we study the possible structures of $mathbb{Q}$-rational preperiodic points which occur under specialization.
We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminan
We demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multipli
The degree-$4$ Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit alg
This paper considers the topological degree of $G$-shifts of finite type for the case where $G$ is a nonabelian monoid. Whenever the Cayley graph of $G$ has a finite representation and the relationships among the generators of $G$ are determined by a
We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.