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Register a new userOn the number of zeros of diagonal cubic forms over finite fields
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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.
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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.
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.
Let $F(boldsymbol{x})$ be a diagonal integer-coefficient cubic form in $min{4,5,6}$ variables. Excluding rational lines if $m=4$, we bound the number of integral solutions $boldsymbol{x}in[-X,X]^m$ to $F(boldsymbol{x})=0$ by $O_{F,epsilon}(X^{3m/4 - 3/2 + epsilon})$, conditionally on an optimal large sieve inequality (in a specific range of parameters) for approximate Hasse-Weil $L$-functions of smooth hyperplane sections $F(boldsymbol{x})=boldsymbol{c}cdotboldsymbol{x}=0$ as $boldsymbol{c}inmathbb{Z}^m$ varies in natural boxes. When $m$ is even, these results were previously established conditionally under Hooleys Hypothesis HW. Our $ell^2$ large sieve approach requires that certain bad factors be roughly $1$ on average in $ell^2$, while the $ell^infty$ Hypothesis HW approach only required the bound in $ell^1$. Furthermore, the large sieve only accepts uniform vectors; yet our initially given vectors are only approximately uniform over $boldsymbol{c}$, due to variation in bad factors and in the archimedean component. Nonetheless, after some bookkeeping, partial summation, and Cauchy, the large sieve will still apply. In an appendix, we suggest a framework for non-diagonal cubics, up to Hessian issues.
Let K be a cubic number field. In this paper, we study the Ramanujan sums c_{J}(I), where I and J are integral ideals in O_{K}. The asymptotic behaviour of sums of c_{J}(I) over both I and J is investigated.
Let Fq be a finite field with q=8 or q at least 16. Let S be a smooth cubic surface defined over Fq containing at least one rational line. We use a pigeonhole principle to prove that all the rational points on S are generated via tangent and secant operations from a single point.
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