Browsing by Author "Sharma, Divyum"

Now showing 1 - 14 of 14

Arithmetic nature of some in nite series and integrals
(TIFR, 2015) Sharma, Divyum
We give an overview of some results on the transcendence nature of the sums of some in nite series. We also give some new results on the transcendence nature of some series involving mildly non-periodic functions and some integrals.
Contributions to a conjecture of Mueller and Schmidt on Thue inequalities
(IAS, 2017-09) Sharma, Divyum
Let F(X, Y ) = s i=0 ai Xri Yr−ri ∈ Z[X, Y ] be a form of degree r = rs ≥ 3, irreducible over Q and having at most s + 1 non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality |F(X, Y )| ≤ h is s2h2/r (1 + log h1/r ). They conjectured that s2 may be replaced by s. Let = max 0≤i≤s max ⎛ ⎝ i−1 w=0 1 ri − rw , s w=i+1 1 rw − ri ⎞ ⎠ . Then we show that s2 may be replaced by max(s log3 s, se ). We also show that if |a0| = |as | and |ai| ≤ |a0| for 1 ≤ i ≤ s − 1, then s2 may be replaced by s log3/2 s. In particular, this is true if ai ∈ {−1, 1}.
Diagonalizable thue equations -- revisited
(2022) Sharma, Divyum
Let r,h∈N with r≥7 and let F(x,y)∈Z[x,y] be a binary form such that F(x,y)=(αx+βy)r−(γx+δy)r, where α, β, γ and δ are algebraic constants with αδ−βγ≠0. We establish upper bounds for the number of primitive solutions to the Thue inequality 0<|F(x,y)|≤h, improving an earlier result of Siegel and of Akhtari, Saradha & Sharma.
Joint distribution in residue classes of the base-q and Ostrowski digital sums
(ARXIV, 2017-10) Sharma, Divyum
Let q be an integer ≥2 and let Sq(n) denote the sum of digits of n in base q. For α=[0;1,m¯¯¯¯¯¯¯¯¯], m≥2, let Sα(n) denote the sum of digits in the Ostrowski α-representation of n. Let m1,m2≥2 be integers with gcd(q−1,m1)=gcd(m,m2)=1. We prove that there exists δ>0 such that for all integers a1,a2, |{0≤n
A Note on Thue Inequalities with Few Coefficients
(Springer, 2016) Sharma, Divyum
Let F(X,Y)=∑i=0saiXriYr−ri∈Z[X,Y] be a form of degree r≥3, irreducible over Q, and having at most s+1 nonzero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality |F(X,Y)|≤h is ≪s2h2/r(1+logh1/r) . They conjectured that s2 may be replaced by s. In this note we show some instances when s2 may be improved.
Number of representations of integers by binary Forms
(TIFR, 2014) Sharma, Divyum
We give improved upper bounds for the number of solutions of the Thue equation F(x; y) = h where F is an irreducible binary form of degree 3:
Number of solutions of cubic Thue inequalities with positive discriminant
(EuDML, 2015) Sharma, Divyum
Let F(X,Y) be an irreducible binary cubic form with integer coefficients and positive discriminant D. Let k be a positive integer satisfying k < ( ( 3 D ) 1 / 4 ) / 2 π . We give improved upper bounds for the number of primitive solutions of the Thue inequality | F ( X , Y ) | ≤ k .
On the coefficient-choosing game
(ARXIV, 2021) Sharma, Divyum
Nora and Wanda are two players who choose coefficients of a degree-d polynomial from some fixed unital commutative ring R. Wanda is declared the winner if the polynomial has a root in the ring of fractions of R and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington, and Zbarsky (2018) to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors. Our techniques there lead us to an adversarial approach to constructing rational polynomials of any prescribed degree (equal to 3 or greater than 8) with no roots in the maximal abelian extension of Q.
On the number of non-zero digits of integers in multi-base representation
(Publicationes Mathematicae Debrecen, 2017-01) Sharma, Divyum
We prove various finiteness theorems for integers having only few non- zero digits in different multi-base representations simultaneously
On the representation of an imaginary quadratic integer in two different bases
(2023) Sharma, Divyum
Let (α,Nα) and (β,Nβ) be two canonical number systems for an imaginary quadratic number field K such that α and β are multiplicatively independent. We provide an effective lower bound for the sum of the number of non-zero digits in the α-adic and β-adic expansions of an algebraic integer γ∈OK which is an increasing function of |γ|. This is an analogue of an earlier result due to Stewart on integer representations
On the representation of an integer in Ostrowski and recurrence numeration systems
(2024-09) Sharma, Divyum
We provide an effective upper bound for positive integers with bounded Hamming weights with respect to both a linear recurrence numeration system and an Ostrowski- numeration system, where is a quadratic irrational. We prove a similar result for the representation of an integer in two different Ostrowski numeration systems
Rational solutions to the Variants of Erdős- Selfridge superelliptic curves
(ARXIV, 2021-05) Sharma, Divyum
Simultaneous rational approximation via Rickert’s integrals
(Elsevier, 2015-01) Sharma, Divyum
Using Rickert’s contour integrals, we give effective lower bounds for simultaneous rational approximations to numbers in the sets Here are integers, is a rational number and at least one of the radicals is irrational in each set. The result is valid for all where denotes the denominator of the approximating rational number.
Thue's inequalities and the hypergeometric method
(ARXIV, 2016-03) Sharma, Divyum
Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape 0<|F(x,y)|≤h, where F(x,y)=(αx+βy)r−(γx+δy)r∈Z[x,y], α, β, γ and δ are algebraic constants with αδ−βγ≠0, and r≥3 and h are integers. As an important application, we pay special attention to the binomial Thue's inequaities |axr−byr|≤c. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.