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I am looking for a reference for the following statement:

For every integer $d$, there exist an integer $k$, such that for all polynomials $P_1, \ldots, P_k$ of degree $d$ there exist integers $N, q, a$, such that all integers $n>N$ satisfying $n\equiv a\pmod{q}$ can be represented as $n=P_1(x_1)+\dots+P_k(x_k)$, where the $x_i$ are integers in the range $\left[\left(\frac{n}{2s}\right)^{1/d}, \left(\frac{2n}{s}\right)^{1/d}\right]$, and $s$ is the sum of the leading coefficients of the $P_i$.

A lot of work has been done by many authors concerning additive problems with different monomials, Waring's problem with general polynomials, and strong localization of variables, but I haven't been able to find a reference for the above statement.

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Morally, any first attack on Waring's problem (e.g. Chapter 2 of Vaughan) works here, but to be rigorous one might modify the argument on the major arcs. If you want to quote an explicit result I suggest Birch; Theorem 1 in Section 7 needs only cosmetic changes to yield the following result:

Theorem (Birch) Let $f(x)\in\mathbb{Z}[x_1,\dotsc,x_k]$ have degree $d$, let $g(x)$ be the leading part. Suppose the singular locus of $g$ has codimension $>(d-1)2^d$ in $\mathbb{C}^k$. Let $\nu\in \mathbb{N}$ and let $\mathscr{B}\subset [-1,1]^k$ be a box of side at most $1$. For some $\delta>0$, the number of integer points $x\in P\mathscr{B}$ with $f(x)=\nu$ is $$ C_{n,f,\mathscr{B}}P^{k-d}+O_{f}(P^{k-d-\delta}) $$ If for each prime $p$ there is $x$ with $f({x})\equiv\nu$ and $\nabla f(x)\not\equiv 0$ mod $p$, and there is a solution $\bar{x}\in \mathscr{B}$ to $g(\bar{x})=1$ with distance $\geq r$ from the real singular locus of $g(\bar{x})=1$, then $C_{n,f,\mathscr{B}}\gg_{f,r} 1$.

Take $\mathscr{B}=\frac{1}{2}\big[(\frac{1}{2s})^{1/d},(\frac{2}{s})^{1/d}\big]^k$ and $P= 2\nu^{1/d}$. If $\nu\gg_f 1$, $k>(d-1)2^d$ and $\sum P_i(x_i)\equiv\nu$ has nonsingular solutions mod $p$ then we deduce that $\sum P_i(x_i)=\nu$ has many solutions in your range.

Why does this follow from Birch's work?

That paper is about some degree $d$ forms $f_1,\dotsc,f_R$. But actually, after the introduction, we can take the $f_i$ to be general degree $d$ polynomials if we replace $f_i$ by its leading part in the following places: the definition of $f^{(i)}_{j_0,\dotsc,j_{d-1}}$ at the start of Section 2; the definition $V(\mu)$ and $V^*$ in formulae (2) and (3) of Section 3; the definition of $I(\mathscr{C};\gamma)$ in formula (9) of Section 5; and throughout Section 6.

In Section 7, Theorem 1 we need to replace $f_i$ by its leading part in the definition of $V^*$ and in the expression $\Phi[f(\bar{x})]$. The theorem above is exactly this in the case $R=1$ of a single polynomial.

The argument really does not change. A little thought to see that Lemma 5.1 is still true, that's it.


Appendix

A hardcore version of the theorem above would be uniform in the lower degree parts of $f$.

Theorem? Let $f$, $g$, $P$ and $\mathscr{B}$ be as above. Suppose the polynomial $P^{-d} f(P x)$ has coefficients bounded by some fixed constant $C$. If the singular locus of $g$ has codimension $>(d-1)2^d$ in $\mathbb{C}^k$, then the number of integer solutions to $f(x)=0$ in $P\mathscr{B}$ is $$ C_{n,f,\mathscr{B}}P^{k-d}+O_{g,C}(P^{k-d-\delta}) $$ If for each $p$ there is a solution over $\mathbb{Q}_p$ with $|\nabla f(x)|_p \geq c(p)$, and there is a real solution $x\in P\mathscr{B}$ with $|\nabla f(P x)| \geq c(\infty)P^d$, then $C_{n,f,\mathscr{B}}\gg_{g,C,c} 1$. (Edit: corrected the condition at $\infty$.)

The condition on $P^{-d} f(P \bar{x})=0$ just means that if $x\in P\mathscr{D}$ then $f(x)$ is not much larger than the leading part $g(x)$, so that the lower degree parts do not dominate too badly. The alleged proof is the same as the theorem above, but you push on through sections 5 and 6 of Birch's paper with $f$ remaining inhomogeneous. I have not checked this. Modern technology might give a neater proof.

I should also mention that everything here would work for systems of several forms, with appropriate changes in the condition on the codimension of the singular locus.

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    $\begingroup$ Thanks, this answer came just in time. Right now I had started to write up the circle method argument $\endgroup$ – Jan-Christoph Schlage-Puchta Feb 9 '18 at 13:35

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