All Questions
Tagged with plane-geometry polynomials
13 questions
45
votes
4
answers
5k
views
Polynomial roots and convexity
A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever ...
12
votes
1
answer
400
views
Probability that random cubic polynomials meet in a square
Let $p_1(x)$ and $p_2(x)$ be cubic polynomials with
random coefficients in $[-1,1]$.
I wanted to compute the probability that $p_1$ and $p_2$
share at least one point within
the square $[-1,1]^2$.
Of ...
- 151k
6
votes
3
answers
497
views
Polynomial threading through a monotone corridor
I have a need to find a polynomial of minimal degree that connects
two points and stays within a given
"corridor," by which I mean an $x$-monotone polygon.
Here is an example:
&...
- 151k
5
votes
2
answers
565
views
Geometry of Level sets of elliptic polynomials in two real variables
Updated:
A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
- 356
5
votes
0
answers
169
views
Plane real curves such that their intersections with lines are hyperbolic
Let $R$ be an (irreducible) plane real algebraic curve (without isolated points).
Consider Zarissky closure of $R$ in ${\mathbb C}P^1$ (as real variety).
Suppose that $\lambda\in R \Rightarrow\...
- 413
4
votes
1
answer
159
views
Best polygonal approximation to a polynomial $\pm$ c
Let a planar region $R$ be defined
by the vertical range bounded by
a polynomial $f(x) \pm c$ with $c>0$ a constant,
and with $x$ varying between the smallest and largest
roots of $f(x)$.
For ...
- 151k
4
votes
1
answer
239
views
Is there any Menelaus-type theorem for polynomials?
Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$.
In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
- 550
4
votes
1
answer
495
views
Cubic curve closest to the given set of points
Assume we are given the set $S$ of $n$ points on the real plane and want to draw a parametrized cubic curve (actually a segment of Bézier spline) with fixed startpoint in such a way, that the ...
- 41
3
votes
1
answer
138
views
A geometric property about certain polynomials in two variables
Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$
where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
- 356
2
votes
1
answer
614
views
Half spaces free of roots of a given polynomial
I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version
This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...
- 356
2
votes
1
answer
194
views
Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$
I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...
- 3,319
1
vote
1
answer
83
views
Showing that a particular area is small
Note: I posted this on math.stackexchange.com earlier (original post here: https://math.stackexchange.com/questions/1471331/showing-that-a-particular-area-is-small), but it received no responses and ...
- 26.9k
0
votes
2
answers
130
views
Algebraic planar curve with precisely $n$ closed components? [closed]
For each integer $n$ I am looking for a real-valued polynomial in two variables, $A_n(x,y)$, such that $A_n(x,y) = 0$ defines a curve with precisely $n$ closed components in the plane $\mathbb{R}^2$. ...
- 17