Decomposition of polynomials and approximate roots

TL;DR

This paper introduces a polynomial decomposition method based on a Euclidean division theorem, providing algorithms to compute approximate roots and determine polynomial decomposability.

Contribution

It presents a novel Euclidean division theorem for polynomials, defines approximate roots, and offers an algorithm for polynomial decomposition and decomposability testing.

Findings

Existence of a Euclidean division theorem for polynomials

Algorithm for computing polynomial decompositions

Application to polynomial decomposability decision

Abstract

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is the approximate d-root of P. Moreover we give an algorithm to compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not.

Equations

$$P(x)=h\circ Q(x)+R(x)$$

$$Q=\sqrt[d]{P}$$

$$P(x)=h\circ Q(x).$$

$$\mathrm{deg}(P-Q^d)<\mathrm{deg}P-\frac{\mathrm{deg}P}{d}.$$

$$\mathcal{S}$$

$$c_{i_1\dots i_{k-1}}=\left(\genfrac{}{}{0ex}{}{d}{i_1,\dots ,i_{k-1}}\right)=\frac{d!}{i_1!\cdots i_{k-1}!(d-i_1-\cdots -i_{k-1})!}.$$

$$P(x)=h\circ Q(x).$$

$$b_1=\frac{a_1}{2},b_2=\frac{a_2-b_1^2}{2},b_3=\frac{a_3-2b_1{b}_2}{2}.$$

$$R(x)=(a_4-2b_1{b}_3-b_2^2)x^2+(a_5-2b_2{b}_3)x.$$

$$\{\begin{array}{l}{a}_4-2b_1{b}_3-b_2^2=0,\\ a_5-2b_2{b}_3=0.\end{array}$$

$$P(x_1,\dots ,x_n)=h\circ Q(x_1,\dots ,x_n).$$

$$\{\begin{array}{l}{a}_6-b_3^2\in K,\\ a_4-2b_1{b}_3-b_2^2=0\text{ in }K[y],\\ a_5-2b_2{b}_3=0\text{ in }K[y].\end{array}$$

Full text

1. Introduction

Let $A$ be an integral domain (i.e. a unitary commutative ring without zero divisors). Our main result is:

Theorem 1**.**

Let $P\in A[x]$ be a monic polynomial. Let $d\geqslant 2$ such that $d$ is a divisor of $\deg P$ and $d$ is invertible in $A$. There exist $h,Q,R\in A[x]$ such that

[TABLE]

with the conditions that

1. (i)

$h,Q$* are monic;* 2. (ii)

$\deg h=d$, $\operatorname{coeff}(h,x^{d-1})=0$, $\deg R<\deg P-\frac{\deg P}{d}$; 3. (iii)

$R(x)=\sum_{i}r_{i}x^{i}$* with $(\deg Q|i\Rightarrow r_{i}=0)$.*

Moreover such $h,Q,R$ are unique.

The previous theorem has a formulation similar to the Euclidian division; but here $Q$ is not given (only its degree is fixed); there is a natural $Q$ (that we will compute, see Corollary 2) associated to $P$ and $d$. Notice also that the decomposition $P(x)=h\circ Q(x)+R(x)$ is not the $Q$-adic decomposition, since the coefficients before the powers $Q^{i}(x)$ belong to $A$ and not to $A[x]$.

Example*.*

Let $P(x)=x^{6}+6x^{5}+6x+1\in\mathbb{Q}[x]$. If $d=6$ we find the following decomposition $P(x)=h\circ Q(x)+R(x)$ with $h(x)=x^{6}-15x^{4}+40x^{3}-45x^{2}+30x-10$, $Q(x)=x+1$ and $R(x)=0$. If $d=3$ we have $h(x)=x^{3}+65$, $Q(x)=x^{2}+2x-4$ and $R(x)=40x^{3}-90x$. If $d=2$ we get $h(x)=x^{2}-\frac{725}{4}$, $Q(x)=x^{3}+3x^{2}-\frac{9}{2}x+\frac{27}{2}$ and $R(x)=-\frac{405}{4}x^{2}+\frac{255}{2}x$.

Theorem 1 will be of special interest when then ring $A$ is itself a polynomial ring. For instance at the end of the paper we give an example of a decomposition of a polynomial in two variables $P(x,y)\in A[x]$ for $A=K[y]$.

The polynomial $Q$ that appears in the decomposition has already been introduced in a rather different context. We denote by $\sqrt[d]{P}$ the approximate $d$-root of $P$. It is the polynomial such that $(\sqrt[d]{P})^{d}$ approximate $P$ in a best way, that is to say $P-(\sqrt[d]{P})^{d}$ has smallest possible degree. The precise definition will be given in section 2, but we already notice the following:

Corollary 2**.**

[TABLE]

We apply these results to another situation. Let $A=K$ be a field and $d\geqslant 2$. $P\in K[x]$ is said to be $d$-decomposable in $K[x]$ if there exist $h,Q\in K[x]$, with $\deg h=d$ such that

[TABLE]

Corollary 3**.**

Let $A=K$ be a field. Suppose that $\operatorname{char}K$ does not divide $d$. $P$ is $d$-decomposable in $K[x]$ if and only if $R=0$ in the decomposition of Theorem 1.

In particular, if $P$ is $d$-decomposable, then $P=h\circ Q$ with $Q=\sqrt[d]{P}$.

After the first version of this paper, M. Ayad and G. Chèze communicated us some references so that we can picture a part of history of the subject. Approximate roots appeared (for $d=2$) in some work of E.D. Rainville [9] to find polynomial solutions of some Riccati type differential equations. An approximate root was seen as the polynomial part of the expansion of $P(x)^{\frac{1}{d}}$ into decreasing powers of $x$. The use of approximate roots culminated with S.S. Abhyankar and T.T. Moh who proved the so-called Abhyankar-Moh-Suzuki theorem in [1] and [2]. For the latest subject we refer the reader to an excellent expository article of P. Popescu-Pampu [8]. On the other hand Ritt’s decompostion theorems (see [10] for example) have led to several practical algorithms to decompose polynomials in one variable into the form $P(x)=h\circ Q(x)$: for example D. Kozen and S. Landau in [6] give an algorithm (refined in [5]) that computes a decomposition in polynomial time. Unification of both subjects starts with P.R. Lazov and A.F. Beardon ([7], [3]) for polynomials in one variable over complex numbers: they notice that the polynomial $Q$ is in fact the approximate $d$-root of $P$.

We define approximate roots in section 2 and prove uniqueness of the decomposition of Theorem 1. Then in section 3 we prove the existence of such decomposition and give an algorithm to compute it. Finally in section 4 we apply these results to decomposable polynomials in one variable and in section 5 to decomposable polynomials in several variables.

2. Approximate roots and proof of the uniqueness

The approximate roots of a polynomial are defined by the following property, [1], [8, Proposition 3.1].

Proposition 4**.**

Let $P\in A[x]$ a monic polynomial and $d\geqslant 2$ such that $d$ is a divisor of $\deg P$ and $d$ is invertible in $A$. There exists a unique monic polynomial $Q\in A[x]$ such that:

[TABLE]

We call $Q$ the approximate $d$-root of $P$ and denote it by $\sqrt[d]{P}$.

Let us recall the proof from [8].

Proof.

Write $P(x)=x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\ldots+a_{n}$ and we search an equation for $Q(x)=x^{\frac{n}{d}}+b_{1}x^{\frac{n}{d}-1}+b_{2}x^{\frac{n}{d}-2}+\cdots+b_{\frac{n}{d}}$. We want $\deg(P-Q^{d})<\deg P-\frac{\deg P}{d}$, that is to say, the coefficients of $x^{n},x^{n-1},\ldots,x^{n-\frac{n}{d}}$ in $P-Q^{d}$ equal zero. By expanding $Q^{d}$ we get the following system of equations:

[TABLE]

where the coefficients $c_{i_{1}\ldots i_{k-1}}$ are the multinomial coefficients defined by the following formula:

[TABLE]

The system ($\mathcal{S}$) being a triangular system, we can inductively compute the $b_{i}$ for $i=1,2,\ldots,\frac{n}{d}$: $b_{1}=\frac{a_{1}}{d}$, $b_{2}=\frac{a_{2}-\binom{d}{2}b_{1}^{2}}{d}$, …Hence the system ($\mathcal{S}$) admits one and only one solution $b_{1},b_{2},\ldots,b_{\frac{n}{d}}$.

Notice that we need $d$ to be invertible in $A$ to compute $b_{i}$. Moreover $b_{i}$ depends only on the first coefficients $a_{1},a_{2},\ldots,a_{\frac{n}{d}}$ of $P$. ∎

Proposition 4 enables us to prove Corollary 2: by condition (ii) of Theorem 1 we know that $\deg(P-Q^{d})<\deg P-\frac{\deg P}{d}$ so that $Q$ is the approximate $d$-root of $P$. Another way to compute $\sqrt[d]{P}$ is to use iterations of Tschirnhausen transformation, see [1] or [8, Proposition 6.3]. We end this section by proving uniqueness of the decomposition of Theorem 1.

Proof.

$Q$ is the approximate $d$-root of $P$ so is unique (see Proposition 4 above). In order to prove the uniqueness of $h$ and $R$, we argue by contradiction. Suppose $h\circ Q+R=h^{\prime}\circ Q+R^{\prime}$ with $R\neq R^{\prime}$; set $r_{i}x^{i}$ to be the highest monomial of $R(x)-R^{\prime}(x)$. From one hand $x^{i}$ is a monomial of $R$ or $R^{\prime}$, hence $\deg Q\nmid i$ by condition (iii) of Theorem 1. From the equality $(h^{\prime}-h)\circ Q=R-R^{\prime}$ we deduce that $i=\deg(R-R^{\prime})$ is a multiple of $\deg Q$ ; that yields a contradiction. Therefore $R=R^{\prime}$, hence $h=h^{\prime}$. ∎

3. Algorithm and proof of the existence

Here is an algorithm to compute the decomposition of Theorem 1.

Algorithm 5**.**

• •

Input.* $P\in A[x]$, $d|\deg P$.*

• •

Output.* $h,Q,R\in A[x]$ such that $P=h\circ Q+R$.*

• •

1st step.* Compute $Q=\sqrt[d]{P}$ by solving the triangular system ($\mathcal{S}$) of Proposition 4. Set $h_{1}(x)=x^{d}$, $R_{1}(x)=0$.*

• •

2nd step.* Compute $P_{2}=P-Q^{d}=P-h_{1}(Q)-R_{1}$. Look for its highest monomial $a_{i}x^{i}$. If $\deg Q|i$ then set $h_{2}(x)=h_{1}(x)+a_{i}x^{\frac{i}{\deg Q}}$, $R_{2}=R_{1}$. If $\deg Q\nmid i$ then $R_{2}(x)=R_{1}(x)+a_{i}x^{i}$, $h_{2}=h_{1}$.*

• •

3thd step.* Set $P_{3}=P-h_{2}(Q)-R_{2}$, look for its highest monomial $a_{i}x^{i}$,…*

• •

…

• •

Final step.* $P_{n}=P-h_{n-1}(Q)-R_{n-1}=0$ yields the decomposition $P=h\circ Q+R$ with $h=h_{n-1}$ and $R=R_{n-1}$.*

The algorithm terminates because the degree of the $P_{i}$ decreases at each step. It yields a decomposition $P=h\circ Q+R$ that verifies all the conditions of Theorem 1: in the second step of the algorithm, and due to Proposition 4 we know that $i<\deg P-\frac{\deg P}{d}$. That implies $\operatorname{coeff}(h_{2},x^{d-1})=0$ and $\deg R_{2}<\deg P-\frac{\deg P}{d}$. Therefore at the end $\operatorname{coeff}(h,x^{d-1})=0$. Of course the algorithm proves the existence of the decomposition in Theorem 1.

4. Decomposable polynomials in one variable

Let $K$ be a field and $d\geqslant 2$. $P\in K[x]$ is said to be $d$-decomposable in $K[x]$ if there exist $h,Q\in K[x]$, with $\deg h=d$ such that

[TABLE]

We refer to [4] for references and recent results on decomposable polynomials in one and several variables.

Proposition 6**.**

Let $A=K$ be a field whose characteristic does not divide $d$. A monic polynomial $P$ is $d$-decomposable in $K[x]$ if and only if $R=0$ in the decomposition $P=h\circ Q+R$.

In view of Algorithm 5 we also get an algorithm to decide whether a polynomial is decomposable or not and in the positive case give its decomposition.

Proof.

If $R=0$ then $P$ is $d$-decomposable. Conversly if $P$ is $d$-decomposable, then there exist $h,Q\in K[x]$ such that $P=h(Q)$. As $P$ is monic we can suppose $h,Q$ monic. Moreover, up to a linear change of coordinates $x\mapsto x+\alpha$, we can suppose that $\operatorname{coeff}(h,x^{d-1})=0$. Therefore $P=h(Q)$ is a decomposition that verifies the conditions of Theorem 1. ∎

Remark*.*

Let $P(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n}$, we first consider $a_{1},\ldots,a_{n}$ as indeterminates (i.e. $P$ is seen as an element of $K(a_{1},\ldots,a_{n})[x]$). The coefficients of $h(x),Q(x)$ and $R(x)=r_{0}x^{k}+r_{1}x^{k-1}+\cdots+r_{k}$ (computed by Proposition 4, the system ($\mathcal{S}$) and Algorithm 5) are polynomials in the $a_{i}$, in particular $r_{i}=r_{i}(a_{1},\ldots,a_{n})\in K[a_{1},\ldots,a_{n}]$, $i=0,\ldots,k$.

Now we consider $a_{1}^{*},\ldots,a_{n}^{*}\in K$ as specializations of $a_{1},\ldots,a_{n}$ and denote by $P^{*}$ the specialization of $P$ at $a_{1}^{*},\ldots,a_{n}^{*}$. Then, by Proposition 6, $P^{*}$ is $d$-decomposable in $K[x]$ if and only if $r_{i}(a_{1}^{*},\ldots,a_{n}^{*})=0$ for all $i=0,\ldots,k$. It expresses the set of $d$-decomposable monic polynomials of degree $n$ as an affine algebraic variety. We give explicit equations in the following example.

Example*.*

Let $K$ be a field of characteristic different from $2$. Let $P(x)=x^{6}+a_{1}x^{5}+a_{2}x^{4}+a_{3}x^{3}+a_{4}x^{2}+a_{5}x+a_{6}$ be a monic polynomial of degree $6$ in $K[x]$ (the $a_{i}\in K$ being indeterminates). Let $d=2$. We first look for the approximate $2$-root of $P(x)$. $\sqrt[2]{P(x)}=Q(x)=x^{3}+b_{1}x^{2}+b_{2}x+b_{3}$. In view of the triangular system ($\mathcal{S}$) we get

[TABLE]

Once we have computed $Q$, we get $h(x)=x^{2}+a_{6}-b_{3}^{2}$. Therefore

[TABLE]

Now $P(x)$ is $2$-decomposable in $K[x]$ if and only if $R(x)=0$ in $K[x]$ that is to say if and only if $(a_{1},\ldots,a_{6})$ satifies the polynomial system of equations in $a_{1},\ldots,a_{5}$:

[TABLE]

5. Decomposable polynomials in several variables

Again $K$ is a field and $d\geqslant 2$. Set $n\geqslant 2$. $P\in K[x_{1},\ldots,x_{n}]$ is said to be $d$-decomposable in $K[x_{1},\ldots,x_{n}]$ if there exist $Q\in K[x_{1},\ldots,x_{n}]$, and $h\in K[t]$ with $\deg h=d$, such that

[TABLE]

Proposition 7**.**

Let $A=K[x_{2},\ldots,x_{n}]$, $P\in A[x_{1}]=K[x_{1},\ldots,x_{n}]$ monic in $x_{1}$. Fix $d$ that divides $\deg_{x_{1}}P$, such that $\operatorname{char}K$ does not divide $d$. $P$ is $d$-decomposable in $K[x_{1},\ldots,x_{n}]$ if and only if the decomposition $P=h\circ Q+R$ of Theorem 1 in $A[x_{1}]$ verifies $R=0$ and $h\in K[t]$ (instead of $h\in K[t,x_{2},\ldots,x_{n}]$).

Proof.

If $P$ admits a decomposition as in Theorem 1 with $R=0$ and $h\in K[t]$ then $P=h\circ Q$ is $d$-decomposable.

Conversly if $P$ is $d$-decomposable in $K[x_{1},\ldots,x_{n}]$ then $P=h\circ Q$ with $h\in K[t]$, $Q\in K[x_{1},\ldots,x_{n}]$. As $P$ is monic in $x_{1}$ we may suppose that $h$ is monic and $Q$ is monic in $x_{1}$. We can also suppose $\operatorname{coeff}(h,t^{d-1})=0$. Therefore $h$, $Q$ and $R:=0$ verify the conditions of Theorem 1 in $A[x]$. As such a decomposition is unique, it ends the proof. ∎

Example*.*

Set $A=K[y]$ and let $P(x)=x^{6}+a_{1}x^{5}+a_{2}x^{4}+a_{3}x^{3}+a_{4}x^{2}+a_{5}x+a_{6}$ be a monic polynomial of degree $6$ in $A[x]=K[x,y]$, with coefficients $a_{i}=a_{i}(y)\in A=K[y]$. In the example of section 4 we have computed the decomposition $P=h\circ Q+R$ for $d=2$ and set $b_{1}=\frac{a_{1}}{2},\quad b_{2}=\frac{a_{2}-b_{1}^{2}}{2},\quad b_{3}=\frac{a_{3}-2b_{1}b_{2}}{2}.$ We found $h(t)=t^{2}+a_{6}-b_{3}^{2}\in A[t]$ and $R(x)=(a_{4}-2b_{1}b_{3}-b_{2}^{2})x^{2}+(a_{5}-2b_{2}b_{3})x\in A[x].$ By Proposition 7 above, we get that $P$ is $2$-decomposable in $K[x,y]$ if and only

[TABLE]

Each line yields a system of polynomial equations in the coefficients $a_{ij}\in K$ of $P(x,y)=\sum a_{ij}x^{i}y^{j}\in K[x,y]$. In particular the set of $2$-decomposable monic polynomials of degree $6$ in $K[x,y]$ is an affine algebraic variety.