= Solution
Let $d=\deg f$ and let $f_d$ be the nonzero highest homogeneous part. Choose one coordinate, after a permutation, such that
$$
f_d(X_1,\ldots,X_{n-1},1)
$$
is not the zero polynomial. A polynomial of degree at most $d$ in each variable cannot vanish on the entire grid $\{0,1,\ldots,d\}^{n-1}$, by induction on the number of variables. Hence there are $a_i\in\{0,1,\ldots,d\}$ such that
$$
f_d(a_1,\ldots,a_{n-1},1)\ne0.
$$
Set
$$
y_i=t_i-a_it_n\quad(1\leq i<n),
\qquad y_n=t_n.
$$
This is given, up to the initial coordinate permutation, by an integer matrix $M$ with determinant $\pm1$ and
$$
\max_{i,j}|M_{ij}|\leq d.
$$
In the inverse coordinates $t_i=y_i+a_iy_n$, the coefficient of $y_n^d$ in $f$ is the nonzero real number $f_d(a_1,\ldots,a_{n-1},1)$. Dividing by it makes the defining equation monic in $y_n$. Thus $A$ is integral over $\mathbb R[y_1,\ldots,y_{n-1}]$ by <linear Noether normalization for a hypersurface>. The <Lying-over theorem> now makes
$$
\operatorname{Spec}A\longrightarrow
\operatorname{Spec}\mathbb R[y_1,\ldots,y_{n-1}]
$$
surjective.
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