Farshid Hajir
University of Massachusetts Amherst
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Featured researches published by Farshid Hajir.
Canadian Journal of Mathematics | 2009
Farshid Hajir
We study the algebraic properties of Generalized Laguerre Polynomials for negative integral valuesof the parameter. For integersr,n ≥ 0, we conjecture thatL ( 1 n r) n (x) = P n j=0 `n j +r n j ´ x j /j! is a Q-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950s by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when r islarge with respect to n ≥ 5. Here we verify it in three situations: (i) when n is large with respect to r, (ii) when r ≤ 8, and (iii) when n ≤ 4. The main tool is the theory of p-adic Newton Polygons.
Compositio Mathematica | 2001
Farshid Hajir; Christian Maire
The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infmR2m. One knows that α0 ≥ 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 ≥ 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.
Experimental Mathematics | 2013
Paul E. Gunnells; Farshid Hajir; Dan Yasaki
Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.
Journal of Physics A | 2015
Anna M. Barry; Farshid Hajir; P. G. Kevrekidis
In this work, we construct suitable generating functions for vortices of alternating signs in the realm of Bose-Einstein condensates. In addition to the vortex-vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend them to larger numbers of vortices. Moreover, particular moment conditions can be derived e.g. about the sum of the squared locations of the vortices for arbitrary vortex numbers. Furthermore, the relevant differential equation can be generalized appropriately in the two-dimensional complex plane and allows the construction e.g. of polygonal vortex ring and multi-ring configurations, as well as ones with rings surrounding a vortex at the center that are again connected to earlier bibliography.
Canadian Journal of Mathematics | 2018
Farshid Hajir; Christian Maire
For a finite abelian p-group A of rank d, we define its (logarithmic) mean exponent to be the base-p logarithm of the d-th root of its cardinality. We study the behavior of the mean exponent of p-class groups in towers of number fields. By combining techniques from group theory with the Tsfasman-Valdut generalization of the Brauer-Siegel Theorem, we construct infinite tamely ramified towers in which the mean exponent of class groups remains bounded. Several explicit examples with p=2 are given. We introduce an invariant M(G) attached to a finitely generated FAb pro-p group G which measures the asymptotic growth of the mean exponent of abelianizations of subgroups of index n with n going to infinity. When G=Gal(L/K), M(G) measures the asymptotic behavior of the mean exponent of class groups in L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.
Pacific Journal of Mathematics | 1996
Farshid Hajir
Journal de Theorie des Nombres de Bordeaux | 2005
Farshid Hajir
Journal of Algebra | 1997
Farshid Hajir
International Mathematics Research Notices | 2005
Wayne Aitken; Farshid Hajir; Christian Maire
Journal of Algebra | 1997
Farshid Hajir