Mean-value conjecture for well-conditioned polynomial tuples

Let $X$ be sufficiently large, and let ${\mathbf f}=(f_1,\ldots,f_t)$ be a well-conditioned $t$-tuple of polynomials with integral coefficients, where $\deg(f_j)=k_j$. Define

$$A_{s,r}(X;{\mathbf f})=\oint |{\mathfrak g}({\boldsymbol\alpha};X)|^{2s}\,{\rm d}{\boldsymbol\alpha},$$

where

$${\mathfrak g}({\boldsymbol\alpha};X)=\sum_{|h|\leqslant X^r}\sum_{1\leqslant z\leqslant X}e\left(h\sum_{j=1}^t f_j(z)\alpha_j\right).$$

Equivalently, $A_{s,r}(X;{\mathbf f})$ counts the solutions of $\sum_{i=1}^{2s}h_i f_j(z_i)=0$ for $1\leqslant j\leqslant t$, with $|h_i|\leqslant X^r$ and $1\leqslant z_i\leqslant X$. Mean-value conjecture for well-conditioned polynomial tuples. For natural numbers $r$, $s$, and $t$, one has

$$A_{s,r}(X;{\mathbf f})\ll X^\varepsilon\left(X^{r(2s-1)+1}+X^{2s(r+1)-tr-k_1-\ldots-k_t}\right).$$

The first term accounts for diagonal solutions, while the second is the expected contribution from the remaining variables and equations. The theorem preceding this conjecture proves the first-term bound when $t\geqslant2s-1$; extending the estimate to larger $s$ is the speculative part and remains open.