We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the theoretical assumption that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.

We give a separation bound for an isolated multiple root x of a square multivariate analytic system f satisfying that the deflation process applied on f and x terminates after only one iteration. When x is only given approximately, we give a numerical criterion for isolating a cluster of zeros of f near x. We also propose a lower bound of the number of roots in the cluster.

Food allergies affect about 4% of the Korean population, and buckwheat allergy is one of the most severe food allergies in Korea. The purpose of the present study was to develop a murine model of IgE-mediated buckwheat hypersensitivity induced by intragastric sensitization. Young female C3H/HeJ mice were sensitized and challenged intragastricly with fresh buckwheat flour (1, 5, 25 mg/dose of proteins) mixed in cholera toxin, followed by intragastric challenge. Anaphylactic reactions, antigen-specific antibodies, splenocytes proliferation assays and cytokine productions were evaluated. Oral buckwheat challenges of sensitized mice provoked anaphylactic reactions such as severe scratch, perioral/periorbital swellings, or decreased activity. Reactions were associated with elevated levels of buckwheat-specific IgE antibodies. Splenocytes from buckwheat allergic mice exhibited significantly greater proliferative responses to buckwheat than non-allergic mice. Buckwheat-stimulated IL-4, IL-5, and INF-γ productions were associated with elevated levels of buckwheat-specific IgE in sensitized mice. In this model, 1 mg and 5 mg dose of sensitization produced almost the same degree of Th2-directed immune response, however, a 25 mg dose showed blunted antibody responses. In conclusion, we developed IgE-mediated buckwheat allergy by intragastric sensitization and challenge, and this model could provide a good tool for future studies.

We present the Macaulay2 package NumericalCertification for certifying roots of square polynomial systems. It employs the interval Krawczyk method and α -theory as main methods for certification. The package works with output data computed in Macaulay2 with no need for external software. Also, our implementation supports the Krawczyk method which uses interval arithmetic.

We develop algorithms for certifying an approximation to a nonsingular solution of a square system of equations built from univariate analytic functions. These algorithms are based on the existence of oracles for evaluating basic data about the input analytic functions. One approach for certification is based on α-theory while the other is based on the Krawczyk generalization of Newton's iteration. We show that the necessary oracles exist for D-finite functions and compare the two algorithmic approaches for this case using our software implementation in SageMath.