Let p>3 and consider a prime power q=ph. We completely characterize permutation polynomials of Fq2 of the type fa,b(X)=X(1+aXq(q−1)+bX2(q−1))∈Fq2X. In particular, using connections with algebraic ...curves over finite fields, we show that the already known sufficient conditions are also necessary.
We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic ...Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne–Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.
This paper is a study of the eigenvalues of a complex square matrix with one variable nondiagonal entry expressed in polar form. Changing the angle of the variable entry while leaving the radius ...fixed generates an algebraic curve; as does the process of fixing an angle and varying the radius. The authors refer to these two curves as eigenvalue orbits and eigenvalue trajectories, respectively. Eigenvalue orbits and trajectories are orthogonal families of curves, and eigenvalue orbits are sets of eigenvalues from matrices with identical Gershgorin regions. Algebraic and geometric properties of both types of curves are examined. Features such as poles, singularities, and foci are discussed.
We consider the recently introduced notion of denominators of Padé-like approximation problems on a Riemann surface. These denominators are related as in the classical case to the notion of ...orthogonality over a contour. We investigate a specific setup where the Riemann surface is a real elliptic curve with two components and the measure of orthogonality is supported on one of the two real ovals. Using a characterization in terms of a Riemann–Hilbert problem, we determine the strong asymptotic behaviour of the corresponding orthogonal functions for large degree. The theory of vector bundles and the non-abelian Cauchy kernel play a prominent role even in this simplified setting, indicating the new challenges that the steepest descent method on a Riemann surface has to overcome.
Let Hd,g,r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in Pr. We denote by Hd,g,rL the union of those components of Hd,g,r whose general ...element is linearly normal and we show that any non-empty Hd,g,rL (d≥g+r−3) is irreducible for an extensive range of triples (d,g,r) beyond the Brill-Noether range. This establishes the validity of a suitably modified assertion of Severi regarding the irreducibility of the Hilbert scheme Hd,g,rL of linearly normal curves for g+r−3≤d≤g+r, r≥3, and g≥2r+3 if d=g+r−3.
A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in C2 is introduced. The structure of irreducible invariant algebraic curves for Liénard dynamical ...systems xt=y, yt=−f(x)y−g(x) with degg=degf+1 is obtained. It is shown that there exist Liénard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Liénard differential system with degf=2 and degg=3 are obtained. All these results seem to be new.
This paper is concerned with cubic Kukles system of the form x˙=−y, y˙=x+λy+a1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7y3, where λ,ai∈R, i=1,2,…,7, under the assumption that the above system has an algebraic ...periodic orbit of degree 2. It is shown that such an algebraic periodic orbit surrounds a center if and only if λ=0. If λ≠0, then it is the unique limit cycle. Moreover, we provide all the possibilities for the global phase portrait in the Poincaré disk of cubic Kukles system with an algebraic periodic orbit of degree 2.
We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials fα,β(x)=x+αxq(q−1)+1+βx2(q−1)+1∈Fq2x, αβ≠0, q even, characterizing all the pairs (α,β)∈Fq22 for which fα,β(x) is a ...permutation of Fq2.
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