Finding a polynomial whose roots are connected to the roots of a different polynomial

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Suppose we have a polynomial function $$f(x) =x^5-4x^4+3x^3-2x^2+5x+1$$ Function $f$ will have 5 roots which can be denoted by $a, b, c, d, e$. I was interested in trying to find a degree 10 polynomial whose roots are given by $abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde$. My idea was that we can relate the coefficients of the degree 10 polynomial to the coefficients of the degree 5 polynomial using Vieta's relations. However, I soon realised that this led to expressions that were extremely difficult to simplify and the method in general, was time-consuming. I was interested in knowing if general techniques exist to solve such problems or if brute is the only way to go about it.

Thanks

Best Answer

Let

  • $g(x) = x^5 f\left(\frac1x\right) = x^5+5x^4-2x^3+3x^2-4x+1$.
  • $S = \{ a,b,c,d,e \}$ be the roots of $f(x)$.
  • $T = \{ \frac1a, \frac1b, \frac1c, \frac1d, \frac1d \}$ be the roots of $g(x)$.
  • For $I \subset S$ and $J \subset T$, let $\lambda_I = \prod_{\lambda \in I}\lambda$ and $\mu_J = \prod_{\mu \in J}\mu$.

The polynomial we seek equals to $\quad\displaystyle\;F(x) \stackrel{def}{=} \prod_{I \subset S,|I| = 3}(x - \lambda_I)$.

Define a similar polynomial for $g$, $\quad\displaystyle\;G(x) \stackrel{def}{=} \prod_{J \subset T,|J| = 2}(x - \mu_J)$.

By Vieta's formula, we have $abcde = -1$, this implies $$F(x) = \prod_{I\subset S,|I|=3} \left(x + \frac{1}{\lambda_{S \setminus I}}\right) = \prod_{J\subset T,|J|=2}(x + \mu_J) = G(-x)$$

The problem comes down to given $g(x)$, how to compute $G(x)$ whose roots are product of distinct pairs of roots of $g(x)$.

It will be hard to relate the coefficients of $g$ and $G$ directly. However, there is a simple relation between the power sums. More precisely, for any $k \in \mathbb{Z}_{+}$, let

  • $P_k(g) \stackrel{def}{=} \sum_{\mu \in T} \mu^k$ be the sum of roots of $f(x)$ raised to power $k$.
  • $P_k(G) \stackrel{def}{=} \sum_{J \subset T,|J|=2} \mu_J^k$ be the sum of roots of $G(x)$ raised to power $k$.

We have $$P_k(G) = \frac12( P_k(g)^2 - P_{2k}(g))\tag{*1}$$

To make following descriptions more generic, let $n = 5$ and $m = \frac{n(n-1)}{2}$.
Define coefficients $\alpha_k, \beta_k$ as follow:

$$g(x) = x^n - \sum\limits_{k=1}^n \alpha_k x^{n-k} \quad\text{ and }\quad G(x) = x^m - \sum\limits_{k=1}^m \beta_k x^{m-k}$$

Following are the steps to compute coefficients $\beta_k$ from coefficients $\alpha_k$ manually.

  1. Compute $P_k(g)$ using Newton's identity for $1 \le k \le 2m$.

$$P_k(g) = \sum_{j=1}^{\min(n,k-1)} \alpha_j P_{k-j}(g) + \begin{cases} k \alpha_k, & k \le n\\ 0, & \text{otherwise}\end{cases} $$

  1. Compute $P_k(G)$ from $P_k(g)$ using $(*1)$.

  2. Compute $\beta_k$ from $P_k(G)$ using Newton's identities again: $$\beta_k = \frac1k\left( P_k(G) - \sum_{j=1}^{k-1} \beta_j P_{k-j}(G) \right)$$

I am lazy, I implement above logic in maxima (the CAS I use) and compute these numbers. The end result is

$$F(x) = x^{10}-2x^9+19x^8-112x^7+82x^6+97x^5-15x^4+58x^3+3x^2+3x+1$$

If one has access to a CAS, there is a quicker way to get the result.

For example, in maxima, one can compute the resultant between $g(t)$ and $g\left(-\frac{x}{t}\right)$ using the command resultant(g(t), g(-x/t), t)). The resultant of two polynomials is essentially the GCD of them over the polynomial ring. It vanishes when and only when the two polynomials share a root. When the resultant between $g(t)$ and $g\left(-\frac{x}{t}\right)$ vanishes, $x$ either equals to $-\mu^2$ for a root $\mu \in T$ or $-\mu\nu$ for some $\mu, \nu$ in $T$.

If one ask maxima to factor output of above command, the result is $$-(x^5+29x^4-34x^3+3x^2+10x+1)F(x)^2$$

The first factor is nothing but $\prod\limits_{\mu \in T}(x + \mu^2)$, this confirm the expression we get for $F(x)$ is the product $\prod\limits_{J \subset T,|J| = 2}(x + \mu_J)$ we desired.

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