Solving Robust Glucose-Insulin Control by Dixon Resultant Computations

We present a symbolic approach towards solving the Bergman three-state minimal patient model of glucose metabolism. Our work first translates the Bergman three-state minimal patient model into the modified control algebraic Riccati equation. Next, the modified control algebraic Ricatti equation is reduced to a system of polynomial equations, and an optimal (minimal) solution of these polynomials is computed using Dixon resultants. We demonstrate the use of our method by reporting on three case studies over glucose metabolism

Multihomogeneous resultant formulae by means of complexes

We provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants arising from Weyman complexes associated to line bundles in products of projective spaces. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. We characterize the systems that admit a purely B\'ezout-type matrix and show a bijection of such matrices with the permutations of the variable groups. We conclude with examples showing the hybrid matrices that may be encountered, and illustrations of our Maple implementation. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing results by Sturmfels-Zelevinsky, and Weyman-Zelevinsky.Comment: 30 pages. To appear: Journal of Symbolic Computatio

New and Old Results in Resultant Theory

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.Comment: 50 pages, 15 figure