We consider certain systems of three linked simultaneous diagonal equations in ten variables with total degree exceeding five. By means a complification argument, we obtain an asymptotic formula for the number integral solutions this system bounded height that resolves associated paucity problem.

A 1976 theorem of Chaitin can be used to show that arbitrarily dense sets of lengths n have a paucity of trivial strings (only a bounded number of strings of length n having trivially low plain Kolmogorov complexities). We use the probabilistic method to give a new proof of this fact. This proof is much simpler than previously published proofs, and it gives a tighter paucity bound.

History: A 3.5-months-old male presented to us with small bowel type of diarrhea and recurrent non-bilious vomiting since birth; jaundice since 2.5 months of age; swelling all over body, clay colored stools, fever, cough and respiratory difficulty since 3 months of age, and refusal to feed from 2 days prior to admission. He also had prolonged bleeding at the site of intramuscular injection. The...

Abstract Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ we show that there is a paucity nondiagonal solutions Diophantine system _j({\mathbf x...