On the number of zeros to the equation $f(x_1)+...+f(x_n)=a$ over finite fields

Let $p$ be a prime, $k$ a positive integer and let $mathbb{F}_q$ be the finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $mathbb F_q$ and $ainmathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of $f(x_1)+cdots+f(x_s)=a$. In this paper, we show that $$sum_{s=1}^{infty}N_{s}(f,0)x^s=frac{x}{1-qx} -frac{x { M_f^{prime}}(x)}{qM_f(x)},$$ where $$M_f(x):=prod_{minmathbb F_q^{ast}atop{S_{f, m} e 0}}Big(x-frac{1}{S_{f,m}}Big)$$ with $S_{f, m}:=sum_{xin mathbb F_q}zeta_p^{{rm Tr}(mf(x))}$, $zeta_p$ being the $p$-th primitive unit root and ${rm Tr}$ being the trace map from $mathbb F_q$ to $mathbb F_p$. This extends Richmans theorem which treats the case of $f(x)$ being a monomial. Moreover, we show that the generating series $sum_{s=1}^{infty}N_{s}(f,a)x^s$ is a rational function in $x$ and also present its explicit expression in terms of the first $2d+1$ initial values $N_{1}(f,a), ..., N_{2d+1}(f,a)$, where $d$ is a positive integer no more than $q-1$. From this result, the theorems of Chowla-Cowles-Cowles and of Myerson can be derived.

On the number of zeros of diagonal quartic forms over finite fields

Let $mathbb{F}_q$ be the finite field of $q=p^mequiv 1pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y inmathbb{F}_q$ with $yinmathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x_1^4+...+x_n^4=c$ and $x_1^4+...+x_{n-1}^4+yx_n^4=0$, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function $sum_{n=1}^{infty}N_n(0)x^n$ is a rational function in $x$ and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions $sum_{n=1}^{infty}N_n(c)x^n$ and $sum_{n=1}^{infty}M_{n+1}(y)x^n$ are rational functions in $x$. We also obtain the explicit expressions of these generating functions. Our result extends Myersons theorem gotten in 1979.

Sums of polynomial-type exceptional units modulo $n$

Let $f(x)inmathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $nge 1$ and $kge 2$. An integer $a$ is called an $f$-exunit in the ring $mathbb{Z}_n$ of residue classes modulo $n$ if $gcd(f(a),n)=1$. In this paper, we use the principle of cross-classification to derive an explicit formula for the number ${mathcal N}_{k,f,c}(n)$ of solutions $(x_1,...,x_k)$ of the congruence $x_1+...+x_kequiv cpmod n$ with all $x_i$ being $f$-exunits in the ring $mathbb{Z}_n$. This extends a recent result of Anand {it et al.} [On a question of $f$-exunits in $mathbb{Z}/{nmathbb{Z}}$, {it Arch. Math. (Basel)} {bf 116} (2021), 403-409]. We derive a more explicit formula for ${mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.

On the number of zeros of diagonal cubic forms over finite fields

Let ${mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, zinmathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+cdots+x_s^3=z$ and $x_1^3+cdots+x_{s-1}^3+yx_s^3=0$, respectively. Gauss proved that if $q=p, pequiv1pmod3$ and $y$ is non-cubic, then $T_3(y)=p^2+frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined by $4p=c^2+27d^2,~cequiv 1 pmod 3$ except for the sign of $d$. In 1978, Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being a non-cubic element of ${mathbb F}_p$. But the sign problem is kept open for the remaining case of $2$ being cubic in ${mathbb F}_p$. In this paper, we solve this sign problem by determining the sign of $d$ when $2$ is cubic in ${mathbb F}_p$. Furthermore, we show that the generating functions $sum_{s=1}^{infty} N_{s}(z) x^{s}$ and $sum_{s=1}^{infty} T_{s}(y)x^{s}$ are rational functions for any $z, yinmathbb F_q^*:=mathbb F_qsetminus {0}$ with $y$ being non-cubic over ${mathbb F}_q$ and also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.

Algebraic properties of summation of exponential Taylor polynomials

Let $nge 1$ be an integer and $e_n(x)$ denote the truncated exponential Taylor polynomial, i.e. $e_{n}(x)=sum_{i=0}^nfrac{x^i}{i!}$. A well-known theorem of Schur states that the Galois group of $e_n(x)$ over $Q$ is the alternating group $A_n$ if $n$ is divisible by 4 or the symmetric group $S_n$ otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials $E_n(x):=e_n(x)+e_{n-1}(x)$. We show that $frac{x^n}{n!}+sum_{i=0}^{n-1}c_ifrac{x^i}{i!}$ with all $c_i (0le ile n-1)$ being integers is irreducible over $Q$ if either $c_0=pm 1$, or $n$ is not a positive power of $2$ but $|c_0|$ is a positive power of 2. This extends another theorem of Schur. We show also that $E_n(x)$ is irreducible if $n otin{2,4}$. Furthermore, we show that ${rm Gal}_{Q}(E_n)$ contains $A_{n}$ except for $n=4$, in which case, ${rm Gal}_{Q}(E_4)=S_3$. Finally, we show that the Galois group ${rm Gal}_{Q}(E_n)$ is $S_n$ if $nequiv 3 pmod 4$, or if $n$ is even and $v_p(n!)$ is odd for a prime divisor of $n-1$, or if $nequiv 1pmod 4$ and $n-2$ equals the product of an odd prime number $p$ which is coprime to $sum_{i=1}^{p-1}2^{p-1-i}i!$ and a positive integer coprime to $p$.

On the $p$-adic properties of Stirling numbers of the first kind

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the $p$-adic valuations of $s(n,k)$. In this paper, by using Washingtons congruence on the generalized harmonic number and the $n$-th Bernoulli number $B_n$ and the properties of $m$-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound of $v_p(s(ap, k))$ with $a$ and $k$ being integers such that $1le ale p-1$ and $1le kle ap$. This infers that for any regular prime $pge 7$ and for arbitrary integers $a$ and $k$ with $5le ale p-1$ and $a-2le kle ap-1$, one has $v_p(H(ap-1,k))<-frac{log{(ap-1)}}{2log p}$ with $H(ap-1, k)$ being the $k$-th elementary symmetric function of $1, frac{1}{2}, ..., frac{1}{ap-1}$. This gives a partial support to a conjecture of Leonetti and Sanna raised in 2017. We also present results on $v_p(s(ap^n,ap^n-k))$ from which one can derive that under certain condition, for any prime $pge 5$, any odd number $kge 3$ and any sufficiently large integer $n$, if $(a,p)=1$, then $v_p(s(ap^{n+1},ap^{n+1}-))=v_p(s(ap^n,ap^n-k))+2$. It confirms partially Lengyels conjecture proposed in 2015.

A certain reciprocal power sum is never an integer

By $(mathbb{Z}^+)^{infty}$ we denote the set of all the infinite sequences $mathcal{S}={s_i}_{i=1}^{infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let $mathcal{S}_n:={s_1, ..., s_n}$ and $H_f(mathcal{S}_n):=sum_{k=1}^{n}frac{1}{f(k)^{s_{k}}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $mathcal{S}$ of positive integers, $H_f(mathcal{S}_n)$ is never an integer if $nge 2$. Now let deg$f(x)ge 2$. Clearly, $0<H_f(mathcal{S}_n)<zeta(2)<2$. But it is not clear whether the reciprocal power sum $H_f(mathcal{S}_n)$ can take 1 as its value. In this paper, with the help of a result of ErdH{o}s, we use the analytic and $p$-adic method to show that for any infinite sequence $mathcal{S}$ of positive integers and any positive integer $nge 2$, $H_f(mathcal{S}_n)$ is never equal to 1. Furthermore, we use a result of Kakeya to show that if $frac{1}{f(k)}lesum_{i=1}^inftyfrac{1}{f(k+i)}$ holds for all positive integers $k$, then the union set $bigcuplimits_{mathcal{S}in (mathbb{Z}^+)^{infty}} { H_f(mathcal{S}_n) | nin mathbb{Z}^+ }$ is dense in the interval $(0,alpha_f)$ with $alpha_f:=sum_{k=1}^{infty}frac{1}{f(k)}$. It is well known that $alpha_f= frac{1}{2}big(pi frac{e^{2pi}+1}{e^{2pi}-1}-1big)approx 1.076674$ when $f(x)=x^2+1$. Our dense result infers that when $f(x)=x^2+1$, for any sufficiently small $varepsilon >0$, there are positive integers $n_1$ and $n_2$ and infinite sequences $mathcal{S}^{(1)}$ and $mathcal{S}^{(2)}$ of positive integers such that $1-varepsilon<H_f(mathcal{S}^{(1)}_{n_1})<1$ and $1<H_f(mathcal{S}^{(2)}_{n_2})<1+varepsilon$.

The 2-adic valuations of Stirling numbers of the first kind

Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_2(s(2^n,k))$ holds for all integers $k$ between 1 and $2^n$. As a corollary, we show that $v_2(s(2^n,2^n-k))=2n-2-v_2(k-1)$ if $k$ is odd and $2le kle 2^{n-1}+1$. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if $kle 2^n$, then $v_2(s(2^n,k)) le v_2(s(2^n,1))$ and $v_2(H(2^n,k))leq -n$, where $H(n,k)$ stands for the $k$-th elementary symmetric functions of $1,1/2,...,1/n$. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

A generalization of a theorem of Nagell

Let $n$ be a positive integer. In 1915, Theisinger proved that if $nge 2$, then the $n$-th harmonic sum $sum_{k=1}^nfrac{1}{k}$ is not an integer. Let $a$ and $b$ be positive integers. In 1923, Nagell extended Theisingers theorem by showing that the reciprocal sum $sum_{k=1}^{n}frac{1}{a+(k-1)b}$ is not an integer if $nge 2$. In 1946, ErdH{o}s and Niven proved a theorem of a similar nature that states that there is only a finite number of integers $n$ for which one or more of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we present a generalization of Nagells theorem. In fact, we show that for arbitrary $n$ positive integers $s_1, ..., s_n$ (not necessarily distinct and not necessarily monotonic), the following reciprocal power sum $$sumlimits_{k=1}^{n}frac{1}{(a+(k-1)b)^{s_{k}}}$$ is never an integer if $nge 2$. The proof of our result is analytic and $p$-adic in character.

The Igusa local zeta functions of superelliptic curves

Let $K$ be a local field and $f(x)in K[x]$ be a non-constant polynomial. The local zeta function $Z_f(s, chi)$ was first introduced by Weil, then studied in detail by Igusa. When ${rm char}(K)=0$, Igusa proved that $Z_f(s, chi)$ is a rational function of $q^{-s}$ by using the resolution of singularities. Later on, Denef gave another proof of this remarkable result. However, if ${rm char}(K)>0$, the question of rationality of $Z_f(s, chi)$ is still kept open. Actually, there are only a few known results so far. In this paper, we investigate the local zeta functions of two-variable polynomial $g(x, y)$, where $g(x, y)=0$ is the superelliptic curve with coefficients in a non-archimedean local field of positive characteristic. By using the notable Igusas stationary phase formula and with the help of some results due to Denef and Z${rm acute{u}}$${rmtilde{n}}$iga-Galindo, and developing a detailed analysis, we prove the rationality of these local zeta functions and also describe explicitly all their candidate poles.