Date modified: Mon 2023-06-05 . 09:40 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Polynomial racing. What is the nature and behavior of real polynomials that share a common intersection? That is, if $n$ real polynomials $p_1,\ldots, p_n$ has graphs that all intersect at $(x,y)=(a,b)$, how do they behave from $x=a^-$ to $x=a^+$ as they go across $x=a$? In particular, the values of $p_i(a^-)$ can be ordered from smallest to largest. How would this ordering change as they become $p_i(a^+)$? === By translating, we can set the common intersection at $(0,0)$, namely they all have common root at $x=0$. So let us take a collection of $n$ **different** real polynomials $p_1, p_2,\ldots , p_n$, each with $0$ as one of its roots, namely $p_i\in x\mathbb R[x]$ for each $i=1,2,\ldots,n$. Since these are nonzero polynomials, and each has finitely many real roots, there exists a small neighborhood $(-\epsilon,\epsilon)$ such that $0$ is the only root to each polynomial $p_1,\ldots, p_n$. In fact, one can choose a small open interval $I$ about $0$ such that $0$ is the only common intersection between any two polynomials among $p_1,\ldots,p_n$. ![[---images/---assets/---icons/question-icon.svg]] Show or convince yourself that this is indeed the case. This means on this small interval $I$, the polynomials $p_1,\ldots,p_n$ can be ordered from smallest to largest on $I_- = I\cap(-\infty,0)$ by their values on $I_-$. And similarly $p_1,\ldots,p_n$ can be ordered from smallest to largest on $I_+ = I\cap(0,\infty)$. If we reindex the polynomials such that $p_1 < p_2 < \cdots < p_n$ on $I_-$, then we get some permutation $\sigma$ such that $p_{\sigma(1)} < p_{\sigma(2)} < \cdots < p_{\sigma(n)}$ on $I_+$. Let us call this permutation $\sigma$ the **racing permutation** of the the collection of polynomials $p_1 , p_2 ,\ldots,p_n \in x\mathbb R[x]$. We can imagine that $n$ polynomials with some initial ordering enters the tunnel at $0$ from $I_-$, and they emerge out from $I_+$ with a new ordering given by the permutation $\sigma$. A curious question we can ask is: Given positive integer $n$, is every permutation $\sigma$ some racing permutation realizable by some set of real polynomials in $x\mathbb R[x]$? ![[---images/---assets/---icons/question-icon.svg]] Let us get used to this definition first. If we have polynomials $x , x^2 ,x^3$, what is the racing permutation of these three polynomials? ![[---images/---assets/---icons/question-icon.svg]] For each of the $6$ possible permutations of $1,2,3$, can you come up with three real polynomials in $x\mathbb R[x]$ such that the permutation is their racing permutation? ![[---images/---assets/---icons/question-icon.svg]] How about for each of the $24$ possible permutations of $1,2,3,4$? Are all of them racing permutations? How many of them are racing polynomials that you can come up with? ///Hint./// There are exactly two permutations that cannot be racing permutations, can you find them? The following exercises might help you. /// As you might have observed already, the behavior of these polynomials near $0$ are largely determined by the **order** of these polynomials, denote as $\operatorname{ord}(p)$, that is, the degree of the **lowest non-zero term**. For example, $\operatorname{ord}(4x^4 - 3x^2 + x^2) = 2$. ![[---images/---assets/---icons/question-icon.svg]] If $p \in x\mathbb R[x]$ is such that $\operatorname{ord}(p)$ is even, how does its graph look like near $0$? And what if $\operatorname{ord}(p)$ is odd? ![[---images/---assets/---icons/question-icon.svg]] If $p_1,p_2 \in x \mathbb R[x]$ are such that $\operatorname{ord}(p_1)\le\operatorname{ord}(p_2)$, what can we say about these polynomials near $0^+$ and $0^-$ in relation to their distance to the horizontal line $y=0$? ![[---images/---assets/---icons/question-icon.svg]] Suppose two polynomials $p_1,p_2 \in x\mathbb R[x]$ are such that $\operatorname{ord}(p_1 - p_2)$ is **even**. What you can say about ordering of $p_1$ and $p_2$ as it goes from $0^-$ to $0^+$? ![[---images/---assets/---icons/question-icon.svg]] Suppose two polynomials $p_1,p_2 \in x\mathbb R[x]$ are such that $\operatorname{ord}(p_1 - p_2)$ is **odd**. What you can say about ordering of $p_1$ and $p_2$ as it goes from $0^-$ to $0^+$? ![[---images/---assets/---icons/question-icon.svg]] Show that the permutation $3142$ is not a a racing permutation. Hint: Take four polynomials from $x\mathbb R[x]$, and without loss we can take one of them to be the zero polynomial (why?) Try to arrive at a contradiction if $3142$ is a racing permutation. What is the other permutation in $S_4$ that is also not a racing permutation? In general, if a permutation $\sigma \in S_n$ contains a pattern that is the above two permutations from $S_4$, then they are also not racing permutations! (And as it turns out, this is necessary and sufficient! But this takes a bit of more work to show.) ![[---images/---assets/---icons/question-icon.svg]] Can you see how this generalizes to real analytic functions with root at $x=0$ as well (and analytic at $x=0$)?