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While browsing through Davenport's lecture notes "Analytic methods for diophantine equations and Diophantine inequalities", near the end of chap 8 I came across the statement that for the equation

$c_1 x_1^k + ... + c_s x_s^k = 0$,

if the $c_i$ are non-zero and, if $k$ is even, the sign of $c_i$ are not all the same, then this equation has infinitely many integer solutions if $s\ge k^2+1$. Davenport then noted that the bound $k^2+1$ is optimal if $k+1$ is prime, and that "for most values of $k$ smaller value than $k^2+1$ will suffice.

Question: What's the current record for $s$ for small $k = 3, 5, 7, 8, ...$ " ? What about conjectural/expected "optimal" value for $s$ for small $k$?

THANKS!

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    $\begingroup$ For $k=3$, see Heath-Brown, The circle method and diagonal cubic forms, royalsocietypublishing.org/doi/10.1098/rsta.1998.0181 also citeseerx.ist.psu.edu/viewdoc/… $\endgroup$ Oct 3, 2021 at 12:57
  • $\begingroup$ Thanks @Myerson. Follow-up question: What is known about lower bound of the number integer solutions say for k=3? $\endgroup$
    – W Sao
    Oct 3, 2021 at 14:52
  • $\begingroup$ I haven't read the Heath-Brown paper carefully, just skimmed part of it, but it seems to me that he says that for $k=3$, $s=4$, you get lines containing infinitely many points and giving a lower bound. But I could be wrong. $\endgroup$ Oct 4, 2021 at 2:12

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In general, what the circle method gives you is an asymptotic formula for the number of solutions to the equation, which will be of the form $C P^{s-k}$ for some non-negative $C$. We are currently (with the help of recent advances on Vinogradov's mean value theorem) able to establish such asymptotic formulae when $s>k^2 + O(k)$ (see Wooley 2012 IMRN). If one doesn't require an asymptotic formula and is instead happy with infinitely many solutions, it is possible to restrict the base set from the integers to smooth numbers (i.e. no large prime divisors). One can then establish a similar asymptotic formula for the number of solutions in as few as $s>k \log k +O(k \log \log k)$ variables. Generally, the expectation is that such an asymptotic formula should hold whenever $s>2k$, and possibly (with a diverging factor $C$) when $s>k$. However, as things stand this is very much pie in the sky territory!

The elephant in the room, however, is the constant $C$. In Davenport's book you can see that $C$ has a representation as the product over the densities of solutions to your equation over all local fields $\mathbb R$ and $\mathbb Q_p$, and this is where the real bottleneck sits. The general rule of thumb is that you can expect a value that is smaller than $k^2+1$ whenever you can prove that there are no $p$-adic obstructions. Understanding the details and particulars of this last question is in general still open, but Mike Knapp and Hemar Godinho have published papers on this, which you could check out.

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  • $\begingroup$ Out of curiosity could you explain the intuition behind the expectation that $s>k$ should be enough? I’m not an expert but somehow I always thought that “square-root cancellation” heuristics would force $s > 2k$ roughly (let’s ignore quadrics!). Of course nobody forces you to blindly put in absolute values when treating the minor arcs, so presumably that’s where I’m inaccurate! $\endgroup$
    – alpoge
    Oct 28, 2021 at 15:48
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    $\begingroup$ Well, there's the heuristics that when an equation of degree $k$ in $s$ variables takes inputs from the range $1 \le x_i \le P$, then each possible value is assumed on average $O(P^{s-k})$ times, and the exponent is positive as soon as $s>k$. The value zero plays a bit of a special role, but Manin's conjecture makes a similar prediction, except occasionally a log that arises for geometric reasons. Moreover, in Chapter 4 of Vaughan's book there is a treatment of the major arcs that gives the expected outcome whenever $s>k$, and also recovers the extra log term predicted by Manin's conjecture. $\endgroup$
    – Somebody
    Oct 28, 2021 at 20:25
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    $\begingroup$ So the $s>2k$ is generally considered to be the limit of the methods, and that stems from the absolute values on the minor arcs you mention. However, there is a paper by Brüdern and Wooley (Crelle 2014) in which they consider pairs of cubics with special configurations of coefficients in which they go below that boundary. Also, we can treat quadratic forms in as few as three variables by the circle method (Heath-Brown, Crelle 1996) $\endgroup$
    – Somebody
    Oct 28, 2021 at 20:30

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