Related papers: The Noether exponent and Jacobi formula

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the…

Let f be a polinomial with coefficients in a finite field F. Let $\Psi : F \to C^{\ast}$ be a non-trivial additive character. In this paper we give bounds for the exponential sums $\sum_{x\in F^n} \Psi (Tr_{F/F_p} (f(x)))$ in some cases…

Let $\theta_3(\tau)=1+2\sum_{\nu=1}^{\infty} q^{\nu^2}$ with $q=e^{i\pi \tau}$ and $\Im (\tau)>0$ denote the Thetanullwert of the Jacobi theta function \[\theta(z|\tau) \,=\,\sum_{\nu=-\infty}^{\infty} e^{\pi i\nu^2\tau + 2\pi i\nu z} \,.\]…

We construct a non-proper set of two variables polynomial maps and study the nowhere vanishing Jacobian condition of the Jacobian conjecture for this set. We obtain some classes of polynomial maps satisfying the 2-dimensional Jacobian…

The discrete counterpart of the problem related to the convergence of the Fourier-Jacobi series is studied. To this end, given a sequence, we construct the analogue of the partial sum operator related to Jacobi polynomials and characterize…

The Weitzenboeck theorem states that the algebra of constants of a linear locally nilpotent derivation of the polynomial algebra K[Z]=K[z_1,...,z_m] in m variables over a field K of characteristic 0 is finitely generated. If m=2n and the…

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

Let $\mathbb{F}_q$ be a finite field with $q=p^e$ elements, where $p$ is a prime and $e\geq 1$ is an integer. Let $\ell<n$ be two positive integers. Fix a monic polynomial $u(x)=x^n +u_{n-1}x^{n-1}+\cdots +u_{\ell+1}x^{\ell+1} \in…

This paper aims to derive explicit and computable error bounds for the asymptotic expansion of the Jacobi polynomials as their degree approaches infinity, using an integral method. The analysis focuses on the outer or oscillatory region of…

Let $K$ be an algebraically closed and complete nonarchimedean field with characteristic $0$ and let $f\in K[z]$ be a polynomial of degree $d\ge 2$. We study the Lyapunov exponent $L(f,\mu)$ of $f$ with respect to an $f$-invariant and…

In this paper we obtained the formula for the number of irreducible polynomials with degree $n$ over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al.(2003)…

Let $f\in\Bbb F_q[X_1,\dots,X_n]$ with $\deg f=d>0$ and let $Z(f)=\{(x_1,\dots,x_n)\in \Bbb F_q^n: f(x_1,\dots,x_n)=0\}$. Ax's theorem states that $|Z(f)|\equiv 0\pmod {q^{\lceil n/d\rceil-1}}$, that is, $\nu_p(|Z(f)|)\ge m(\lceil…

We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to the classical weight function for the Jacobi polynomials together with point masses at both…

Noether's problem is classical and very important problem in algebra. It is an intrinsically interesting problem in invariant theory, but with far reaching applications in the sutdy of moduli spaces, PI-algebras, and the Inverse problem of…

The $T$-adic exponential sum of a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the…

We will give several reduction theorems for Noether's problem.

Polynomials with all the coefficients in $\{ 0,1\}$ and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in $\{ -1,1\}$ are called Littlewood polynomials. By exploiting an algorithm developed…

In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi equations in two…

In this note, we consider a complete intersection $X=\{x\in \mathbb{R}^n : f_1(x)= \ldots = f_m(x)=0\}, n>m$ and study its Euclidean distance degree in terms of the mixed volume of the Newton polytopes. We show that if the Newton polytopes…

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For…