Let (K, v) be a valued field and $\phi\in K[x]$ be any key polynomial for a residue-transcendental extension w of v to K(x). In this article, using the ϕ-Newton polygon of a polynomial $f\in K[x]$ (with respect to w), we give a lower bound for the degree of an irreducible factor of f. This generalizes the result given in Jakhar and Srinivas (On the irreducible factors of a polynomial II, J. Algebra 556 (2020), 649–655).

Let $K={\mathbb {Q}}(\theta )$ be an algebraic number field with $\theta $ satisfying a monic irreducible polynomial $f(x)$ of degree n over ${\mathbb {Q}}.$ The polynomial $f(x)$ is said to be monogenic if $\{1,\theta ,\ldots ,\theta ^{n-1}\}$ is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones [‘A brief note on some infinite families of monogenic polynomials’, Bull. Aust. Math. Soc. 100 (2019), 239–244] conjectured that if $n\ge 3$, $1\le m\le n-1$, $\gcd (n,mB)=1$ and A is a prime number, then the polynomial $x^n+A (Bx+1)^m\in {\mathbb {Z}}[x]$ is monogenic if and only if $n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ is square-free. We prove that this conjecture is true.

We study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$. We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$, where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$, where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$. Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$, and thus $\Delta(\mathcal{P}_n)$, for all squarefree $n\equiv11\pmod{24}$.

We prove an improvement on Schmidt’s upper bound on the number of number fields of degree n and absolute discriminant less than X for $6\leq n\leq 94$ . We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].

Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial $D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation $D(x) \equiv 0 \pmod {p}$ has a solution.

We prove a new irreducibility result for polynomials over ${\mathbb Q}$ and we use it to construct new infinite families of reciprocal monogenic quintinomials in ${\mathbb Z}[x]$ of degree $2^n$ .

We give effective finiteness results for the power values of polynomials with coefficients composed of a fixed finite set of primes; in particular, of Littlewood polynomials.

We prove Lawton's conjecture about the upper bound on themeasure of the set on the unit circle on which a complex polynomial with a bounded number of coefficients takes small values. Namely, we prove that Lawton's bound holds for polynomials that are not necessarily monic. We also provide an analogous bound for polynomials in several variables. Finally, we investigate the dependence of the bound on the multiplicity of zeros for polynomials in one variable.

In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.

In this paper, we give a general explicit form of Cassels’ $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

We give an explicit formula for the resultant of Chebyshev polynomials of the first, second, third, and fourth kinds. We also compute the resultant of modified cyclotomic polynomials.

We investigate the numbers of complex zeros of Littlewood polynomials $p\left( z \right)$ (polynomials with coefficients {−1, 1}) inside or on the unit circle $\left| z \right|\,=\,1$, denoted by $N\left( p \right)$ and $U\left( p \right)$, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for $N\left( p \right)$, $U\left( p \right)$ for polynomials $p\left( z \right)$ of these types. We show that if $n\,+\,1$ is a prime number, then for each integer $k,\,0\,⩽\,k\,⩽\,n-1$, there exists a Littlewood polynomial $p\left( z \right)$ of degree $n$ with $N\left( p \right)\,=\,k$ and $U\left( p \right)\,=\,0$. Furthermore, we describe some cases where the ratios $N\left( p \right)/n$ and $U\left( p \right)/n$ have limits as $n\,\to \,\infty $ and find the corresponding limit values.

In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group ${{A}_{4}}$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log \,\left| P \right|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log \,\left| P \right|$ for possibly different ${{P}^{'}}\text{s}$), multiple Mahler measure (involving products of $\log \,\left| P \right|$ for possibly different ${{P}^{'}}\text{s}$), and higher Mahler measure (involving ${{\log }^{k}}\,\left| P \right|$).

We prove a necessary and sufficient condition on the list of nonzero integers ${{u}_{1}},...,{{u}_{k}}$, $k\,\ge \,2$, under which a monic polynomial $f\,\in \,\mathbb{Z}\left| x \right|$ is expressible by a linear form ${{u}_{1}}\,{{f}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{u}_{k}}\,{{f}_{k}}$ in monic polynomials ${{f}_{1}},...,\,{{f}_{k}}\,\in \,\mathbb{Z}\left| x \right|$. This condition is independent of $f$. We also show that if this condition holds, then the monic polynomials ${{f}_{1}},...,\,{{f}_{k}}$ can be chosen to be irreducible in $\mathbb{Z}\left[ x \right]$.

We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes, and V. Trevisan.

A monic polynomial in ${\mathbf{F} }_{q} [t] $ of degree $n$ over a finite field ${\mathbf{F} }_{q} $ of odd characteristic can be written as the sum of two irreducible monic elements in ${\mathbf{F} }_{q} [t] $ of degrees $n$ and $n- 1$ if $q$ is larger than a bound depending only on $n$. The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable $x$ with coefficients in ${\mathbf{F} }_{q} [t] $.

In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0in K[x]and the integer m≥2is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x)with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha $ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.