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The PDE is $$ (\partial_x+s\partial_y+\frac{sx-y}2\partial_z)h=g+B(s)h. $$ The characteristic equation is $$ \frac{dy}{dx}=s,\quad \frac{dz}{dx}=\frac{sx-y}2,\quad \frac{dh}{dx}=g+B(s)h=g+O(\epsilon)h …

I saw this question, but I think the answer didn't fully address what I want to know about it: Nash's paper on parabolic equations. It says almost everything developed later in elliptic and paraboli …

This is a straightforward application of degree theory. First of all, by induction you can reduce to the case when $p\in(0,1)^n$. Then, note that the a linear homotopy between $f$ and the identity m …

It may also be helpful to search for "Waring's problem in finite fields". For a bound derived from discrete Fourier analysis, see https://dl.dropboxusercontent.com/u/27883775/math%20notes/analytic-nt. …

By the compactness of the torus, there is a uniform radius of convergence $r>0$ working for every point. You can extend $f$ to complex variables and use Cauchy's formula to find $|\partial^k f|\le Ck! …

Integrating by parts, using a cutoff function if necessary, we have $$ \int_{B(r)} |\nabla u|^2\ll \int_{B(r)} a_{ij}\partial_iu\partial_ju=\int_{B(R)} uLu+O(u|\nabla u|)\ll \int_{B(R)} |u\nabla u|. …

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the …