Effective approximation to complex algebraic numbers by quadratic numbers
We establish an effective improvement on the Liouville inequality for approximation to complex non-real algebraic numbers by quadratic complex algebraic numbers.
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arXiv subjects
Prajeet Bajpai
Publications and source records attributed to Prajeet Bajpai.
An annotated bibliography for comparative prime number theory
The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory (through mid-2024) together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained.
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Effective $S$-unit Equations Beyond 3 Terms : Newman's Conjecture
We show how to effectively solve 5-term $S$-unit equations when the set of primes $S$ has cardinality at most 3, and use this to provide an explicit answer to an old question of D.J. Newman on representations of integers as sums of $S$-units.
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Effective Approximation to Complex Algebraic Numbers by Algebraic Numbers of Bounded Degree
We establish the first effective improvements on the Liouville inequality for approximation to complex non-real algebraic numbers by complex algebraic numbers of degree at most 4.
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Arithmetic Progressions in Squarefull Numbers
We answer a number of questions of Erdős on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the $abc$-conjecture of Masser and Oesterlé.
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Effective Methods for Norm-Form Equations
While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to address the case of three variables over totally complex and Galois number fields. In this paper, we extend his results to effectively resolve several new classes of norm-form equations. In particular, we completely and effectively settle the question of norm-form equations over totally complex Galois sextic fields.
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A note on pencils of norm-form equations
We find all solutions to the parametrized family of norm-form equations $x^3-(t^3-1)y^3+3(t^3-1)xy+(t^3-1)^2 = \pm 1$ studied by Amoroso, Masser and Zannier. Our proof relies upon an appeal to lower bounds for linear forms in logarithms and various elementary arguments.
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Equivariant division
Let C be a non-empty finite set, and Gamma a subgroup of the symmetric group S(C). Given a bijection f:A cross C to B cross C, the problem of Gamma-equivariant division is to find a quotient bijection h:A to B respecting whatever symmetries f may have under the action of S(A) cross S(B) cross Gamma. Say that Gamma is fully cancelling if this is possible for any f, and finitely cancelling if it is possible providing A,B are finite. Feldman and Propp showed that a permutation group is finitely cancelling just if it has a globally fixed point. We show here that a permutation group is fully cancelling just if it is trivial. This sheds light on the fact that all known division algorithms that eschew the Axiom of Choice depend on fixing an ordering for the elements of C.
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