Let 𝔽q{\mathbb{F}_{q}} be the finite field of q=pk{q=p^{k}} elements with p being a prime and let k be a positive integer. For any y,z∈𝔽q{y,z\in\mathbb{F}_{q}}, let Ns⁢(z){N_{s}(z)} and Ts⁢(y){T_{s}(y)} denote the numbers of zeros of x13+⋯+xs3=z{x_{1}^{3}+\cdots+x_{s}^{3}=z} and x13+⋯+xs-13+y⁢xs3=0{x_{1}^{3}+\cdots+x_{s-1}^{3}+yx_{s}^{3}=0}, respectively. Gauss proved that if q=p{q=p}, p≡1(mod3){p\equiv 1~{}(\bmod~{}3)} and y is non-cubic, then T3⁢(y)=p2+12⁢(p-1)⁢(-c+9⁢d),T_{3}(y)=p^{2}+\frac{1}{2}(p-1)(-c+9d), where c and d are uniquely determined by 4⁢p=c2+27⁢d2{4p=c^{2}+27d^{2}} and c≡1(mod3){c\equiv 1~{}(\bmod~{}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 𝔽p{\mathbb{F}_{p}}. But the sign problem is kept open for the remaining case of 2 being cubic in 𝔽p{\mathbb{F}_{p}}. In this paper, we solve this sign problem by determining the sign of d when 2 is cubic in 𝔽p{\mathbb{F}_{p}}. Furthermore, we show that the generating functions ∑s=1∞Ns⁢(z)⁢xs{\sum_{s=1}^{\infty}N_{s}(z)x^{s}} and ∑s=1∞Ts⁢(y)⁢xs{\sum_{s=1}^{\infty}T_{s}(y)x^{s}} are rational functions for any z,y∈𝔽q*:=𝔽q∖{0}{z,y\in\mathbb{F}_{q}^{*}:=\mathbb{F}_{q}\setminus\{0\}} with y being non-cubic over 𝔽q{\mathbb{F}_{q}}, and we also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.

中文翻译：

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关于有限域上对角立方形式的零个数

设𝔽q{\ mathbb {F} _ {q}}为q = pk {q = p ^ {k}}元素的有限域，其中p为素数，k为正整数。对于任何y，z∈𝔽q{y，z \ in \ mathbb {F} _ {q}}，令Ns⁢（z）{N_ {s}（z）}和Ts⁢（y）{T_ {s} （y）}表示x13 +⋯+ xs3 = z {x_ {1} ^ {3} + \ cdots + x_ {s} ^ {3} = z}和x13 +⋯+ xs-13 +y⁢的零个数xs3 = 0 {x_ {1} ^ {3} + \ cdots + x_ {s-1} ^ {3} + yx_ {s} ^ {3} = 0}。高斯证明如果q = p {q = p}，p≡1（mod3）{p \ equiv 1〜{}（\ bmod〜{} 3）}并且y是非三次的，则T3 T（y）= p2 +12⁢（p-1）⁢（-c +9⁢d），T_ {3}（y）= p ^ {2} + \ frac {1} {2}（p-1）（-c + 9d），其中c和d由4⁢p= c2 +27⁢d2{4p = c ^ {2} + 27d ^ {2}}和c≡1（mod3）{c \ equiv 1〜{}唯一确定（\ bmod〜{} 3）}，但d的符号除外。1978年，对于2是𝔽p{\ mathbb {F} _ {p}}的非立方元素的情况，Chowla，Cowles和Cowles确定了d的符号。但是符号问题对于𝔽p{\ mathbb {F} _ {p}}中2为立方的其余情况保持开放。在本文中，我们通过确定𝔽p{\ mathbb {F} _ {p}}中2为三次的d的正负号来解决该正负号问题。此外，我们证明了生成函数∑s =1∞Ns⁢（z）⁢xs{\ sum_ {s = 1} ^ {\ infty} N_ {s}（z）x ^ {s}}和∑s = 1∞Ts⁢（y）⁢xs{\ sum_ {s = 1} ^ {\ infty} T_ {s}（y）x ^ {s}}是任何z，y∈𝔽q*：=𝔽q∖的有理函数{0} {z，y \ in \ mathbb {F} _ {q} ^ {*}：= \ mathbb {F} _ {q} \ setminus \ {0 \}}，其中y在𝔽q上非三次\ mathbb {F} _ {q}}，我们还给出了它们的显式表达式。这扩展了Myerson定理和Chowla，Cowles和Cowles的定理。我们证明了生成函数∑s =1∞Ns⁢（z）⁢xs{\ sum_ {s = 1} ^ {\ infty} N_ {s}（z）x ^ {s}}和∑s =1∞ Ts⁢（y）⁢xs{\ sum_ {s = 1} ^ {\ infty} T_ {s}（y）x ^ {s}}是任何z，y∈𝔽q*：=𝔽q∖{0的有理函数} {z，y \ in \ mathbb {F} _ {q} ^ {*}：= \ mathbb {F} _ {q} \ setminus \ {0 \}}，其中y在𝔽q{\ mathbb上是非立方的{F} _ {q}}，我们还给出了它们的明确表达。这扩展了Myerson定理和Chowla，Cowles和Cowles的定理。我们证明了生成函数∑s =1∞Ns⁢（z）⁢xs{\ sum_ {s = 1} ^ {\ infty} N_ {s}（z）x ^ {s}}和∑s =1∞ Ts⁢（y）⁢xs{\ sum_ {s = 1} ^ {\ infty} T_ {s}（y）x ^ {s}}是任何z，y∈𝔽q*：=𝔽q∖{0的有理函数} {z，y \ in \ mathbb {F} _ {q} ^ {*}：= \ mathbb {F} _ {q} \ setminus \ {0 \}}，其中y在𝔽q{\ mathbb上是非立方的{F} _ {q}}，我们还给出了它们的明确表达。这扩展了Myerson定理和Chowla，Cowles和Cowles的定理。