Considering Research Denominators

Uncommon Denominators

1974 Emerging Infectious Diseases • www.cdc.gov/eid • Vol. 13, No. 12, December 2007 “O will have greater skill for getting the breath of life to spring from bronze more fl uidly .... But as for you, Roman, remember to impose your power upon nations. Your art is to decree the rules of peace, to spare the vanquished and subdue the vainglorious,” advised Virgil in the Aeneid, placing himself in t...

Universal Denominators of Hilbert

The denominator of the Hilbert series of a finitely generated R-module M does not always divide the denominator of the Hilbert series of R. For this reason, we define the universal denominator. The universal denominator of a module M is the least common multiple of the denominators of the Hilbert series of all submodules of M. The universal denominator behaves nicely with respect to short exact...

Denominators of Egyptian Fractions

Denominators of Igusa Class Polynomials

— In [22], the authors proved an explicit formula for the arithmetic intersection number (CM(K).G1) on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field K. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. One of the...

Treating Small Denominators without Kam

where v : R/Z → R is the potential, α ∈ R \ Q is the frequency, and θ ∈ R is the phase. The most important example is given by the almost Mathieu operator, when v(x) = 2 cos(2πx). There are two classical small denominator problems associated with this model. First, in the regime of small λ , the system is close to constant coefficients and thus, it is natural to attempt to prove reducibility to...