Burchnall–Chaundy polynomials for matrix ODOs and Picard–Vessiot theory

Burchnall and Chaundy showed that if two ordinary differential operators (ODOs) P, Q with analytic coefficients commute then there exists a polynomial f(λ, μ) with complex coefficients such that f(P, Q) = 0, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Our matrices have entries in a differential field K, whose field of constants C is algebraically closed and of zero characteristic. We restrict to the case of order one operators P, with invertible leading coefficient. We define a new differential elimination tool, the matrix differential resultant. We use it to compute the BC-polynomial f of a pair of commuting MODOs, and we also prove that it has constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem PY = λY, QY = μY to have a solution. Techniques from differential algebra and Picard–Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs C[P, Q] and a finite product of rings of irreducible algebraic curvesPID2021-124473NB-I0

Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory

Burchnall and Chaundy showed that if two ODOs $P$, $Q$ with analytic coefficients commute there exists a polynomial $f(\lambda ,\mu)$ with complex coefficients such that $f(P,Q)=0$, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Matrices have entries in a differential field $K$, whose field of constants $C$ is algebraically closed and of zero characteristic. We restrict to the case of order one operators $P$, with invertible leading coefficient. A new differential elimination tool is defined, the matrix differential resultant. It is used to compute the BC-polynomial $f$ of a pair of commuting MODOs and proved to have constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem $PY=\lambda Y \ , \ QY=\mu Y$ to have a solution. Techniques from differential algebra and Picard-Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs $C[P,Q]$ and a finite product of rings of irreducible algebraic curves

Some remarks on the hyperelliptic moduli of genus 3

In 1967, Shioda \cite{Shi1} determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in \cite{Shi1} are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field $k$, $\ch k \neq 2, 3, 5, 7$. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.Comment: arXiv admin note: text overlap with arXiv:1209.044

Akns Hierarchy, Self-Similarity, String Equations and the Grassmannian

In this paper the Galilean, scaling and translational self--similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non--scaling limit analysis of the one--matrix model are shown to correspond to the Galilean self--similarity condition for this hierarchy. We describe, in terms of the initial data for the zero--curvature 1--form of the AKNS hierarchy, the moduli space of these self--similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal--Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura--Hirota rational solutions of the NLS equation. An explicit 1--parameter family of Galilean self--similar solutions of the AKNS equation and the associated solution to the NLS equation is determined.Comment: 25 pages in AMS-LaTe

Trypanosoma cruzi Adjuvants Potentiate T Cell-Mediated Immunity Induced by a NY-ESO-1 Based Antitumor Vaccine

Immunological adjuvants that induce T cell-mediate immunity (TCMI) with the least side effects are needed for the development of human vaccines. Glycoinositolphospholipids (GIPL) and CpGs oligodeoxynucleotides (CpG ODNs) derived from the protozoa parasite Trypanosoma cruzi induce potent pro-inflammatory reaction through activation of Toll-Like Receptor (TLR)4 and TLR9, respectively. Here, using mouse models, we tested the T. cruzi derived TLR agonists as immunological adjuvants in an antitumor vaccine. For comparison, we used well-established TLR agonists, such as the bacterial derived monophosphoryl lipid A (MPL), lipopeptide (Pam3Cys), and CpG ODN. All tested TLR agonists were comparable to induce antibody responses, whereas significant differences were noticed in their ability to elicit CD4+ T and CD8+ T cell responses. In particular, both GIPLs (GTH, and GY) and CpG ODNs (B344, B297 and B128) derived from T. cruzi elicited interferon-gamma (IFN-γ) production by CD4+ T cells. On the other hand, the parasite derived CpG ODNs, but not GIPLs, elicited a potent IFN-γ response by CD8+ T lymphocytes. The side effects were also evaluated by local pain (hypernociception). The intensity of hypernociception induced by vaccination was alleviated by administration of an analgesic drug without affecting protective immunity. Finally, the level of protective immunity against the NY-ESO-1 expressing melanoma was associated with the magnitude of both CD4+ T and CD8+ T cell responses elicited by a specific immunological adjuvant

The resultant on compact Riemann surfaces

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.Comment: 44 page