Composite lacunary polynomials and the proof of a conjecture of Schinzel

Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then h(x)\in \C[x] must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by Erd\"os, Schinzel had proved this in the special cases $g(x)=x^d$; however that method does not extend to the general case. Here we prove the full Schinzel's conjecture (actually in sharper form) by a completely different method. Simultaneously we establish an "algorithmic" parametric description of the general decomposition $f(x)=g(h(x))$, where $f$ is a polynomial with a given number of terms and $g,h$ are arbitrary polynomials. As a corollary, this implies for instance that a polynomial with $l$ terms and given coefficients is non-trivially decomposable if and only if the degree-vector lies in the union of certain finitely many subgroups of $\Z^l$.Comment: 9 page

Linear independence of Gamma values in positive characteristic

We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear independence relations over the field of algebraic functions are consequences of the known relations of Anderson and Thakur arising from the functional equations of the Gamma function.Comment: 51 page

Zero Order Estimates for Analytic Functions

The primary goal of this paper is to provide a general multiplicity estimate. Our main theorem allows to reduce a proof of multiplicity lemma to the study of ideals stable under some appropriate transformation of a polynomial ring. In particular, this result leads to a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. On the other hand, it allows to establish an improvement of Nesterenko's conditional result on solutions of systems of differential equations. We also deduce, under some condition on stable varieties, the optimal multiplicity estimate in the case of generalized Mahler's functional equations, previously studied by Mahler, Nishioka, Topfer and others. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969), and simultaneously it gives a counterpart in the case of functional systems for an important unconditional result of Nesterenko (1977) concerning linear differential systems. In summary, we provide a new universal tool for transcendental number theory, applicable with fields of any characteristic. It opens the way to new results on algebraic independence, as shown in Zorin (2010).Comment: 42 page

Permutable entire functions satisfying algebraic differential equations

It is shown that if two transcendental entire functions permute, and if one of them satisfies an algebraic differential equation, then so does the other one.Comment: 5 page

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

Zero estimates on group varieties II

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46615/1/222_2005_Article_BF01388605.pd

Linear spaces on the intersection of cubic hypersurfaces

Upper bounds for the number of variables necessary to imply the existence of an m -dimensional linear variety on the intersection of r cubic hypersurfaces over local and global fields are given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41633/1/605_2006_Article_BF02349626.pd

Conserved BK Channel-Protein Interactions Reveal Signals Relevant to Cell Death and Survival

The large-conductance Ca2+-activated K+ (BK) channel and its β-subunit underlie tuning in non-mammalian sensory or hair cells, whereas in mammals its function is less clear. To gain insights into species differences and to reveal putative BK functions, we undertook a systems analysis of BK and BK-Associated Proteins (BKAPS) in the chicken cochlea and compared these results to other species. We identified 110 putative partners from cytoplasmic and membrane/cytoskeletal fractions, using a combination of coimmunoprecipitation, 2-D gel, and LC-MS/MS. Partners included 14-3-3γ, valosin-containing protein (VCP), stathmin (STMN), cortactin (CTTN), and prohibitin (PHB), of which 16 partners were verified by reciprocal coimmunoprecipitation. Bioinformatics revealed binary partners, the resultant interactome, subcellular localization, and cellular processes. The interactome contained 193 proteins involved in 190 binary interactions in subcellular compartments such as the ER, mitochondria, and nucleus. Comparisons with mice showed shared hub proteins that included N-methyl-D-aspartate receptor (NMDAR) and ATP-synthase. Ortholog analyses across six species revealed conserved interactions involving apoptosis, Ca2+ binding, and trafficking, in chicks, mice, and humans. Functional studies using recombinant BK and RNAi in a heterologous expression system revealed that proteins important to cell death/survival, such as annexinA5, γ-actin, lamin, superoxide dismutase, and VCP, caused a decrease in BK expression. This revelation led to an examination of specific kinases and their effectors relevant to cell viability. Sequence analyses of the BK C-terminus across 10 species showed putative binding sites for 14-3-3, RAC-α serine/threonine-protein kinase 1 (Akt), glycogen synthase kinase-3β (GSK3β) and phosphoinositide-dependent kinase-1 (PDK1). Knockdown of 14-3-3 and Akt caused an increase in BK expression, whereas silencing of GSK3β and PDK1 had the opposite effect. This comparative systems approach suggests conservation in BK function across different species in addition to novel functions that may include the initiation of signals relevant to cell death/survival

L-Ilf3 and L-NF90 Traffic to the Nucleolus Granular Component: Alternatively-Spliced Exon 3 Encodes a Nucleolar Localization Motif

Ilf3 and NF90, two proteins containing double-stranded RNA-binding domains, are generated by alternative splicing and involved in several functions. Their heterogeneity results from posttranscriptional and posttranslational modifications. Alternative splicing of exon 3, coding for a 13 aa N-terminal motif, generates for each protein a long and short isoforms. Subcellular fractionation and localization of recombinant proteins showed that this motif acts as a nucleolar localization signal. Deletion and substitution mutants identified four arginines, essential for nucleolar targeting, and three histidines to stabilize the proteins within the nucleolus. The short isoforms are never found in the nucleoli, whereas the long isoforms are present in the nucleoplasm and the nucleoli. For Ilf3, only the posttranslationally-unmodified long isoform is nucleolar, suggesting that this nucleolar targeting is abrogated by posttranslational modifications. Confocal microscopy and FRAP experiments have shown that the long Ilf3 isoform localizes to the granular component of the nucleolus, and that L-Ilf3 and L-NF90 exchange rapidly between nucleoli. The presence of this 13 aminoacid motif, combined with posttranslational modifications, is responsible for the differences in Ilf3 and NF90 isoforms subcellular localizations. The protein polymorphism of Ilf3/NF90 and the various subcellular localizations of their isoforms may partially explain the various functions previously reported for these proteins