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Forms in prime variables and differing degrees
Number Theory · Submitted 10 May 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove an asymptotic formula for the number of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$, whose main term is positive if (i) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution over the $p$-adic units $\mathbb{U}_p$ for all primes $p$, and (ii) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution in the open cube $(0,1)^n$.
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The abstract, sentence by sentence
Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. Suppose that $\boldsymbol{F}$ is a nonsingular system and $n\ge D^2 4^{D+6}R^5$. We prove an asymptotic formula for the number of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$, whose main term is positive if (i) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution over the $p$-adic units $\mathbb{U}_p$ for all primes $p$, and (ii) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution in the open cube $(0,1)^n$. This can be viewed as a smooth local-global principle for $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ with differing degrees. It follows that, under (i) and (ii), the set of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ is Zariski dense in the set of its solutions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. | background 0.98 | 0.09 | 0.01 | asserted 0.70 |
| s1 | Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. | background 0.91 | 0.55 | 0.02 | asserted 0.63 |
| s2 | Suppose that $\boldsymbol{F}$ is a nonsingular system and $n\ge D^2 4^{D+6}R^5$. | background 0.70 | 0.16 | 0.03 | asserted 0.62 |
| s3 | We prove an asymptotic formula for the number of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$, whose main term is positive if (i) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution over the $p$-adic units $\mathbb{U}_p$ for all primes $p$, and (ii) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution in the open cube $(0,1)^n$. | result 0.91 | 0.96 | 0.92 | asserted 1.00 |
| s4 | This can be viewed as a smooth local-global principle for $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ with differing degrees. | interpretation 1.00 | 0.55 | 0.10 | asserted 0.57 |
| s5 | It follows that, under (i) and (ii), the set of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ is Zariski dense in the set of its solutions. | interpretation 0.75 | 0.88 | 0.25 | asserted 1.00 |
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Original abstract
Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. Suppose that $\boldsymbol{F}$ is a nonsingular system and $n\ge D^2 4^{D+6}R^5$. We prove an asymptotic formula for the number of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$, whose main term is positive if (i) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution over the $p$-adic units $\mathbb{U}_p$ for all primes $p$, and (ii) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution in the open cube $(0,1)^n$. This can be viewed as a smooth local-global principle for $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ with differing degrees. It follows that, under (i) and (ii), the set of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ is Zariski dense in the set of its solutions.
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References
How the literature uses it
2 arXiv papers cite this in 11 sentences
Papers citing it · 4 sentences without a confident label
Uses its method or tool 1 paper
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The circle method begins with [equation] where [equation] The cube $(0,1]^R$ is partitioned into major arcs $𝔐$ and minor arcs $𝔪$ as in [ref] .
Multiple Gauss sums · Application to the Birch–Goldbach problem · 0.93
Compares with it 1 paper
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This improves [ref] , which required $s≥D^2 4^D+6R^5$.
Multiple Gauss sums · Application to the Birch–Goldbach problem · 0.99
Cites it as background 1 paper
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More recently it has been observed that under suitable rank and local conditions diophantine systems also admit solutions among the primes, and in fact asymptotic formulae may be obtained for the number of prime solutions [ref] .
On almost primes solutions to forms of odd degrees in many variables · Introduction. · 0.81
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