arXiv subjects

N. Saradha

Publications and source records attributed to N. Saradha.

7 recordsLinked to original sources

arXiv2025

Explicit and Mixed Estimates for Thue inequalities with few coefficients

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were written giving an upper bound for the number of solutions of the above inequality as $\ll c(r,s,h)$ where $c(r,s,h)$ is an explicit function of $r,s$ and $h.$ Invariably, the absolute constant involved in $\ll$ has been left undetermined. In this paper, following Bombieri, Schmidt and Mueller, we give three different upper bounds which are explicit in every aspect.

math.NT↗

arXiv2023

Rational solutions to the Variants of Erdős- Selfridge superelliptic curves

For the superelliptic curves of the form $$ (x+1) \cdots(x+i-1)(x+i+1)\cdots (x+k)=y^\ell$$ with $x,y \in \mathbb{Q}$, $y\neq 0$, $k \geq 3$, $1\leq i\leq k$, $\ell \geq 2,$ a prime, Das, Laishram, Saradha, and Edis showed that the superelliptic curve has no rational points for $\ell\geq e^{3^k}$. In fact, the double exponential bound, obtained in these papers is far from reality. In this paper, we study the superelliptic curves for small values of $k$. In particular, we explicitly solve the above equation for $4 \leq k \leq 8.$

math.NT↗

arXiv2022

Diagonalizable Thue Equations -- revisited

Let $r,h\in\mathbb{N}$ with $r\geq 7$ and let $F(x,y)\in \mathbb{Z}[x ,y]$ 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 $αδ-βγ\neq 0$. We establish upper bounds for the number of primitive solutions to the Thue inequality $0<|F(x, y)| \leq h$, improving an earlier result of Siegel and of Akhtari, Saradha & Sharma.

math.NT↗

arXiv2017

On interlacing of zeros of certain family of modular forms

Let $k=12 m(k)+s \ge 12$ for $s\in \{0,4,6,8,10,14\}$, be an even integer and $f$ be a normalised modular form of weight $k$ with real Fourier coefficients, written as $$ f=E_k+\sum_{j=1}^{m(k)}a_jE_{k-12j}Δ^j. $$ Under suitable conditions on $a_j$ (rectifying an earlier result of Getz), we show that all the zeros of $f$, in the standard fundamental domain for the action of ${\bf SL}(2,\mathbb Z)$ on the upper half plane, lies on the arc $A:= \left\{ e^{i θ} : \fracπ{2} \le θ\le \frac{2π}{3} \right\}$. Further, extending a result of Nozaki, we show that for certain family $\{f_k\}_k$ of normalised modular forms, the zeros of $f_k$ and $f_{k+12}$ interlace on $A^\circ:= \left\{ e^{i θ} : \fracπ{2} < θ< \frac{2π}{3} \right\}$.

math.NT↗

arXiv2017

Thue's inequalities and the hypergeometric method

Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape $0<|F(x, y)| \leq h$, where $F(x , y) =(αx + βy)^r -(γx + δy)^r \in \mathbb{Z}[x ,y]$, $α$, $β$, $γ$ and $δ$ are algebraic constants with $αδ-βγ\neq 0$, and $r \geq 3$ and $h$ are integers. As an important application, we pay special attention to the binomial Thue's inequaities $|ax^r - by^r| \leq c$. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.

math.NT↗

arXiv2016

Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

Let $F(X,Y)=\sum\limits_{i=0}^sa_iX^{r_i}Y^{r-r_i}\in\mathbb{Z}[X,Y]$ be a form of degree $r=r_s\geq 3$, irreducible over $\mathbb{Q}$ and having at most $s+1$ non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality \[ |F(X,Y)|\leq h \] is $\ll s^2h^{2/r}(1+\log h^{1/r})$. They $\textit{conjectured}$ that $s^2$ may be replaced by $s$. Let \[ Ψ= \max_{0\leq i\leq s} \max\left( \sum_{w=0}^{i-1}\frac{1}{r_i-r_w},\sum_{w= i+1}^{s}\frac{1}{r_w-r_i}\right). \] Then we show that $s^2$ may be replaced by $\max(s\log^3s, se^Ψ)$. We also show that if $|a_0|=|a_s|$ and $|a_i|\leq |a_0|$ for $1\leq i\leq s-1$, then $s^2$ may be replaced by $s\log^{3/2}s$. In particular, this is true if $a_i\in\{-1,1\}$.

math.NT↗

arXiv2011

On a conjecture of Pomerance

We say that k is a P-integer if the first phi(k) primes coprime to k form a reduced residue system modulo k. In 1980 Pomerance proved the finiteness of the set of P-integers and conjectured that 30 is the largest P-integer. We prove the conjecture assuming the Riemann Hypothesis. We further prove that there is no P-integer between 30 and 10^11 and none above 10^3500.

math.NT↗