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. 



Reference Request: Waring's problem for different polynomials
Morally, any first attack on Waring's problem (e.g. Chapter 2 of [Vaughan](http://dx.doi.org/10.1017/CBO9780511470929)) 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](http://www.jstor.org/stable/2414232); 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.


