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The answer is 'no', because polynomial mappings with polynomial inverses preserve volumes up to a constant multiple. To see why this property holds, suppose that $p:\mathbb{R}^d\to\mathbb{R}^d$ is a polynomial mapping with polynomial inverse $q:\mathbb{R}^d\to\mathbb{R}^d$. Then $p$ and $q$ extend to $\mathbb{C}^d$ as polynomial maps with polynomial ...
Francesco Polizzi's idea is enough to solve the problem: Lemma. Let $f = ax^3 + bx^2 + cx + d \in \mathbf Z_{\geq 0}[x]$ nonconstant with $d > 0$, such that $f(1)$ is a prime $p > 2$. Then $f$ is irreducible in $\mathbf Z[x]$. Proof. Suppose $f = gh$ for $g,h \in \mathbf Z[x]$; we must show that $g \in \{\pm 1\}$ or $h \in \{\pm 1\}$. First assume $\... 4 I will concentrate on the case of odd$q$and$\xi,-\eta$being squares, but the solution should be extendable to the remaining cases. It is convenient to use the language of characters sums in order to compute$N$. To reduce your point counting problem to a character sum problem, the following observation is useful: $$\#\{ x \in \mathbb{F}_q : ax^2+bx+c=0\} ... 3 Integral points on elliptic can often be computed routinely. In Question 1, the curve can be rewritten as$$Y^2 = 5184 + 432 X -12X^2 + X^3,$$where$X:=6n$and$Y:=72y$. SageMath computes: sage: EllipticCurve([0,-12,0,432,5184]).integral_points() [(0 : 72 : 1), (21 : 135 : 1)] So, the only integer solution is$(n,y) = (0,1)$. For Question 2 with$N=4$, we ... 2 You can get all the way up to$\binom{n-1}{2}+1 = g+1$, where$g$is the genus. This is the maximal number of connected components a real curve of genus$g$can have, so this is optimal. To do this, I will use Viro's patchworking method. Itenberg and Viro already give an example of how to use patchworking to build a plane curve with$g+1$connected ... 2 An illustration for one of the examples in the answer by Robert Bryant. It is supposed to convey the feeling of something extremely rigid, unyielding and inflexible. Image of the square$[-1,1]\times[-1,1]$under the map$(x,y)\mapsto(x-y^2-2x^2y-x^4,y+x^2)$(composite of$(x,y)\mapsto(x-y^2,y)$with$(x,y)\mapsto(x,y+x^2)$). 1 Equivalently, you want to interpolate the points$(i, p_i)$,$i = m \ldots n$where$p_i$is the$i$'th prime. The prime$k$-tuples conjecture implies that for each integer$k > 2$and each$d$from$1$to$k-1$, there are infinitely many$m$such that with$n=m+k$the minimum degree of the interpolating polynomial is$d$. The conjecture says that$(p_{m+...