We define an integer polynomial as polynomial that has only integer coefficients. Here I am only interested in polynomials in two variables.
- $P = 5x^4 + 7 x^3y^4 + 4y$
Note that each polynomial P defines a curve by considering the set of points where it evaluates to zero. We will speak about this curve.
The circle can be described by
- $x^2 + y^2 -1 = 0$
We say two polynomials $P,Q$ are touching in point $(a,b)$ if $P(a,b) = Q(a,b) = 0$ and the tangent at $(a,b)$ is the same. Or more geometrically, the curves of $P$ and $Q$ are not crossing.
(The Figure was created with IPE - drawing editor.)
We also need a further technical condition. For this let $D$ be a ''small enough'' disk around $(a,b)$. Then $Q$ and $P$ define two regions indicated green and yellow. Those regions must be interior disjoint. Without this condition for $P = y-x^3$ and $Q=y$ the point $(0,0)$ would be a touching point as well. See also the right side of the figure. (I know that I am not totally precise here, but I don't want to be too formal, so that I can reach a wide audience.) (Thanks for the comment from Jeppe Stig Nielsen.)
- $P = y - x^2$ (Parabola)
- $Q = y$ ($x$-axis)
They touch at the origin $(0,0)$.
Does there exist two integer polynomials $P,Q$ that touch in an irrational point $(a,b)$? (It would be fine for me if either $a$ or $b$ is irrational)
Many thanks for answers and comments. Till