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

Uniparental Disomy and Genomic Imprinting in Humans

Uniparental disomy (UPD), the inheritance of both homologues from one chromosome from the same parent, was first proposed in 1980 by Erik Engel [1] to be a potential cause of congenital developmental defects in hymans. First hints from the premolecular era towards its existence came from instances where a pericentric inversion was present on one homologue in a parent and on both in one offspring [2] and where there was transmission of an interhomologous Robertsonian translocation (of chromosome 22) from a healthy mother to healthy offspring [3-4]. In mice, UPD was experimentally produced by crossing two mice lines with different Robertsonian translocations both involving the same chromosome [for 2 review see ref. 5]. Through this approach, it was possible to define imprinted regions, chromosomes and chromosomal segments for which either maternal or paternal or both types of uniparental disomy led to phenotypic abnormalities. The latter are explained by genomic imprinting, the differential silencing of a gene or genes from one of the parents (the mother or the father) during any stage of embryogenesis or later in life. If, for example, the maternal homologue of a given gene is imprinted (and hence only the paternal allele is active), maternal UPD would lead to loss of the active allele and thus might cause consequences due to loss of functio

Triples of Positive Integers with the same Sum and the same Product

It is proved that for every k there exist k triples of positive integers with the same sum and the same product

Screening for UBE3A gene mutations in a group of Angelman syndrome patients selected according to non-stringent clinical criteria

Abstract.: The Angelman syndrome (AS) is caused by genetic abnormalities affecting the maternal copy of chromosome region 15q12. Until recently, the molecular diagnosis of AS relied on the detection of either a deletion at 15q11-13, a paternal uniparental disomy (UPD) for chromosome 15 or imprinting mutations. A fourth class of genetic defects underlying AS was recently described and consists of mutations of the UBE3A gene. The vast majority of mutations reported so far are predicted to cause major disruptions at the protein level. It is unclear whether mutations with less drastic consequences for the gene product could lead to milder forms of AS. We report on our results obtained by screening 101 clinically diagnosed AS patients for mutations in the UBE3A gene. Non-stringent clinical criteria were purposely applied for inclusion of AS patients in this study. The mutation search was carried out by single-strand conformation polymorphism (SSCP), and SSCP/restriction fragment length polymorphism (RFLP) analyses and revealed five novel UBE3A gene mutations as well as three different polymorphisms. All five mutations were detected in patients with typical features of AS and are predicted to cause frameshifts in four cases and the substitution of a highly conserved residue in the fifth. The results we obtained add to the as yet limited number of reports concerning UBE3A gene mutations. Important aspects that emerge from the data available to date is that the four classes of genetic defects known to underlie AS do not appear to cover all cases. The genetic defect underlying approximately 10% of AS cases, including some familial cases, remains unknow

The irrationality of a number theoretical series

Denote by $\sigma_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k=\sum_{n\geq 1}\frac{\sigma_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the case $k=3$

4q32-q35 and 6q16-q22 are valuable candidate regions for split hand/foot malformation

On the basis of the Human Cytogenetic Database, a computerized catalog of the clinical phenotypes associated with cytogenetically detectable human chromosome aberrations, we collected from the literature 102 cases with chromosomal aberrations and split hand/foot malformation or absent fingers/toes. Statistical analysis revealed a highly significant association (P<0.001) between the malformation and the chromosomal bands 4q32-q35, 5q15, 6q16-q22 and 7q11.2-q22 (SHFM1). Considering these findings, we suggest additional SHFM loci on chromosome 4q, 6q and probably 5q. The regions 4q and 6q have already been discussed in the literature as additional SHFM loci. We now show further evidence. In the proposed regions, there are interesting candidate genes such as, on 4q: HAND2, FGF2, LEF1 and BMPR1B; on 5q: MSX2, FLT4, PTX1 and PDLIM7; and on 6q: SNX3, GJA1, HEY2 and Tbx18.European Journal of Human Genetics advance online publication, 18 February 2009; doi:10.1038/ejhg.2009.11

More Discriminants with the Brezing-Weng Method

The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number