Fermat Type Research Articles

Transcendental entire solutions of several Fermat type PDEs and PDDEs with two complex variables

This article is concerned with the description of transcendental entire solutions of several Fermat type functional equations in ℂ2 concerning partial differential-difference equations and partial differential equations. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we establish some results about the existence and the forms of entire solutions for such equations, which are some improvements and generalizations of the previous theorems. Moreover, some examples are given illustrating that our results are precise to some extent.

On diagonal equations over finite fields

Let Fq be a finite field with q=pn elements. In this paper, we study the number of solutions of equations of the form a1x1d1+…+asxsds=b with xi∈Fpti, where ai,b∈Fq and ti|n for all i=1,…,s. In our main results, we employ results on quadratic forms to give an explicit formula for the number of solutions of diagonal equations with restricted solution sets satisfying certain natural restrictions on the exponents. As a consequence, we present conditions for the existence of solutions. In the second part of the paper, we focus on the case t1=…=ts=n. A classic well-known result from Weil yields a bound for such number of solutions. In the case d1=…=ds, we present necessary and sufficient conditions for the number of solutions of a diagonal equation being maximal and minimal with respect to Weil's bound. In particular, we completely characterize maximal and minimal Fermat type curves. We also discuss further questions concerning equations and present some open problems.

On the transcendental solutions of Fermat type delay-differential and $c$-shift equations with some analogous results

In this paper, we mainly investigate on the finite order transcendental entire solutions of two Fermat types delay-differential and one Fermat type c-shift equations, as these types were not considered earlier. Our results improve those of [13] in some sense. In addition, we also extend some recent results obtained in [18]. A handful number of examples have been provided by us to justify our certain assertion as and when required.

Difference radical in terms of shifting zero and applications to the Stothers-Mason theorem

In this paper, we study the shifting zeros with its heights and an analogue to difference radical. We focus on the Stothers-Mason theorem by using falling factorials. As applications, we discuss the difference version of the Fermat type functional equations. Some examples are given.

Exact forms of entire solutions for Fermat type partial differential equations in C^2

This article studies the existence and the exact form of entire solutions of several Fermat type partial differential equations in \(\mathbb{C}^2\), by utilizing the Nevanlinna theory of meromorphic functions in several complex variables. We obtain results about the existence and form of transcendental entire solutions with finite order for some variations of Fermat type functional equations. Our results are extensions and generalizations of the previous theorems by Xu and Cao [29,30], Liu and Dong [19].
 For more information see https://ejde.math.txstate.edu/Volumes/2021/18/abstr.html

Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

This article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type α ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 1 z 1 , z 2 / ∂ z 2 2 n 1 + f 2 z 1 + c 1 , z 2 + c 2 m 1 = 1 α ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 2 z 1 , z 2 / ∂ z 2 2 n 2 + f 1 z 1 + c 1 , z 2 + c 2 m 2 = 1 and ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 2 + f 2 z 1 + c 1 , z 2 + c 2 2 = 1 ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 2 + f 1 z 1 + c 1 , z 2 + c 2 2 = 1 . Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.

Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables

This paper is concerned with description of the existence and the forms of entire solutions of several second-order partial differential-difference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.

Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables

<abstract><p>By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate some properties of the transcendental entire solutions for several systems of partial differential difference equations of Fermat type, and obtain some results about the existence and the forms of transcendental entire solutions of the above systems, which improve and generalize the previous results given by Cao, Gao, Liu <sup>[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b24">24</xref>,<xref ref-type="bibr" rid="b39">39</xref>]</sup>. Some examples are given show that there exist some significant differences in the forms of transcendental entire solutions with finite order of the systems of equations with between several complex variables and a single complex variable.</p></abstract>

On the entire solutions for several partial differential difference equations (systems) of Fermat type in $\mathbb{C}^2$

By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we will establish some theorems about the existence and the forms of entire solutions for several partial differential difference equations (systems) of Fermat type with two complex variables such as $ f(z)^2+\left[f(z+c)+\frac{\partial f}{\partial z_1}+\frac{\partial f}{\partial z_2}\right]^2 = 1 $ and $ \left\{ \begin{aligned} &amp;f_1(z)^2+\left[f_2(z+c)+\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}\right]^2 = 1, \\ &amp;f_2(z)^2+\left[f_1(z+c)+\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}\right]^2 = 1, \end{aligned}\right. $ which are some extensions and generalizations of the previous theorems given by Xu and Cao ^{[<span class="xref"><a href="#b29" ref-type="bibr">29</a></span>,<span class="xref"><a href="#b30" ref-type="bibr">30</a></span>]}, Xu, Liu and Li ^{[<span class="xref"><a href="#b28" ref-type="bibr">28</a></span>]}, and Liu, Yang ^{[<span class="xref"><a href="#b18" ref-type="bibr">18</a></span>,<span class="xref"><a href="#b19" ref-type="bibr">19</a></span>,<span class="xref"><a href="#b20" ref-type="bibr">20</a></span>]}. Moreover, we give some examples to explain that our results are precise to some extent.

On ternary Diophantine equations of signature(p,p,2)over number fields

Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

Mirror Symmetry for Nonabelian Landau-Ginzburg Models

We consider Landau-Ginzburg models stemming from groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group $G^\star$, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials., we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials.

Restrictions on meromorphic solutions of Fermat type equations

AbstractThe Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

The existence and forms of solutions for some Fermat-type differential-difference equations

The main aim of this article is to investigate the existence and the forms of solutions for several complex differential-difference equations of Fermat-type. Our results about the existence and the forms of solutions for these Fermat type equations are great improvement of the previous theorems given by Liu, Yang, Cao, Zhang. Moreover, it is a very satisfactory fact that in some examples explicit solutions are given.

Entire Solutions of Some Differential–Difference Equations

We use Nevanlinna theory to study the existence of entire solutions with finite order of the Fermat type differential–difference equations. Moreover, we consider the common solutions of a pair of differential and difference equations and give an application in the uniqueness problem of the entire functions.

Curves on weighted K3 surfaces of degree two with symmetric Weierstrass semigroups

Let $C$ be a curve on the weighted projective plane $\mathbf{P}(1, 1, 4)$ which is of Fermat type or almost Fermat type. We construct K3 surfaces which are double covers of $\mathbf{P}(1, 1, 4)$ and which contain pointed curves with symmetric Weierstrass semigroups which are double covers of $C$.

Meromorphic solutions of Fermat type partial differential equations

The main purpose of this paper is to describe the pseudo-primeness of the meromorphic solutions of the Fermat type functional equation F(uz1,⋯,uzn)+uk=1, where F(z1,⋯,zn) is a nonconstant homogeneous polynomial with degree m in n variables z1,⋯,zn and k is a positive integer. In addition, we give a simple proof and strengthening of a theorem which is related to the Fermat type functional equation f3+g3=1 in C2 by improving the condition so that fz2 and gz1 have the same zeros (ignoring multiplicities).

Weighted sharing and uniqueness for the shifts of meromorphic functions

In this paper, we deal with the uniqueness problem for $q$-shifts of meromorphic functions. The results obtained in this paper are the $q$-shift analogues of theorems given by Yang and Zhang [Non-existence of meromorphic solution of a Fermat type functional equation, Aequationes Math. 76(2008), 140-150], Chen, Chen and Li [Uniqueness of difference operators of meromorphic functions, J. Ineq. Appl. 2012(2012), Art 48].

Exact Computation of the Special Geometry for Calabi–Yau Hypersurfaces of Fermat Type

We continue to develop our method for effectively computating the special K\"ahler geometry on the moduli space of Calabi-Yau manifolds. We generalize it to all polynomial deformations of Fermat hypersurfaces.

On Meromorphic Solutions of Functional Equations of Fermat Type

Take complex numbers $a_j,b_j$, $(j=0,1,2)$ such that $c\neq0$ and {\rm rank} ( {ccc} a_{0} & a_{1} & a_{2} b_{0} & b_{1} & b_{2} )=2. We show that if the following functional equation of Fermat type \left\{a_{0}f(z)+a_{1}f(z+c)+a_{2}f'(z)\right\}^3+\left\{b_{0}f(z)+b_{1}f(z+c)+b_{2}f'(z)\right\}^3=e^{\alpha z+\beta} has meromorphic solutions of finite order, then it has only entire solutions of the form $f(z)=Ae^{\frac{\alpha z+\beta}{3}}+Ce^{Dz},$ which generalizes the results in {19} and {14}.

Exact evaluation of second moments associated with some families of curves over a finite field

Let Fq be the finite field with q elements. Given an N-tuple Q∈FqN, we associate with it an affine plane curve CQ over Fq. We consider the distribution of the quantity q−#Cq,Q where #Cq,Q denotes the number of Fq-points of the affine curve CQ, for families of curves parameterized by Q. Exact formulae for first and second moments are obtained in several cases when Q varies over a subset of FqN. Families of Fermat type curves, Hasse–Davenport curves and Artin–Schreier curves are also considered and results are obtained when Q varies along a straight line.