Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups

The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, \emph{the Galois-like groups}) and the triviality of the exponent lattice of a polynomial are investigated. The \bbbq\emph{-trivial} pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.Comment: 19 pages,2 figure

Continued fractions and irrationality exponents for modified engel and pierce series

An Engel series is a sum of reciprocals of a non-decreasing sequence (xn) of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence with the same property. Given an arbitrary rational number, we show that there is a family of Engel series which when added to it produces a transcendental number ? whose continued fraction expansion is determined explicitly by the corresponding sequence (xn), where the latter is generated by a certain nonlinear recurrence of second order. We also present an analogous result for a rational number with a Pierce series added to or subtracted from it. In both situations (a rational number combined with either an Engel or a Pierce series), the irrationality exponent is bounded below by (3 + ?5)/2, and we further identify infinite families of transcendental numbers ? whose irrationality exponent can be computed precisely. In addition, we construct the continued fraction expansion for an arbitrary rational number added to an Engel series with the stronger property that x2j divides xj+1 for all j

p38 Mitogen-Activated Protein Kinase and Liver X Receptor-α Mediate the Leptin Effect on Sterol Regulatory Element Binding Protein-1c Expression in Hepatic Stellate Cells

Leptin, a key hormone in regulating energy homeostasis, is mainly produced by adipocytes. Cogent evidence indicates a unique role of leptin in the promotion of liver fibrosis. Hepatic stellate cell (HSC) activation is a pivotal step in the process of liver fibrosis. Sterol regulatory element binding protein (SREBP)-1c, a critical transcription factor for lipid synthesis and adipocyte differentiation, functions as a key transcription factor in inhibition of HSC activation. SREBP-1c is highly expressed in quiescent HSCs and downregulated upon HSC activation. The aim of this study is to examine the effect of leptin on SREBP-1c gene expression in HSCs in vitro and in vivo and elucidate the underlying mechanisms. The results of the present study demonstrated that leptin strongly inhibited SREBP-1c expression in HSCs in vivo and in vitro. p38 MAPK was involved in leptin regulation of SREBP-1c expression in cultured HSCs. Leptin-induced activation of p38 MAPK led to the decreases in liver X receptor (LXR)-α protein level, activity and its binding to the SREBP-1c promoter, which caused the downregulation of SREBP-1c expression. Moreover, leptin inhibition of SREBP-1c expression via p38 MAPK increased the expression of alpha1(I) collagen in HSCs. Our results might provide new insights into the mechanisms of the unique role of leptin in the development of liver fibrosis and might have potential implications for clarifying the molecular mechanisms underlying liver fibrosis in diseases in which circulating leptin levels are elevated such as nonalcoholic steatohepatitis, type 2 diabetes mellitus and alcoholic cirrhosis