Solutions of the equation $a_{n}+(a_{n-1}+\cdots(a_{2}+(a_{1}+x^{r_{1}})^{r_{2}}\cdots)^{r_{n}}=b\,x$

The equation

$$a_{n}+(a_{n-1}+\cdots(a_{2}+(a_{1}+x^{r_{1}})^{r_{2}}\cdots)^{r_{n}}=b\,x$$

(1)

frequently emerges in problems concerning the qualitative theory of vector fields. In these problems, the variables $(b,a,r)$, belonging to $\mathbb{R}_{>0}\times\mathbb{R}^{2n}$, are treated as free parameters. The goal is to estimate the number of real solutions $x$ to this equation, uniformly with respect to the parameters.

For instance, consider a smooth planar vector field $X$ whose phase portrait has a hyperbolic polycycle $\Gamma$ with $n$ saddle points $s_{1},\ldots,s_{n}$ and let $X_{\lambda}$ be a smooth family of vector fields which unfolds $X$. We fix a local transverse section $\Sigma$ (see picture below) such that $\Sigma\cap\Gamma=\{p\}$ and consider the first return map $\pi:(\Sigma,p)\rightarrow(\Sigma,p)$, which depends on the parameter $\lambda$. The limit cycles bifurcating from $\Gamma$ are related to the isolated fixed points of $\pi$.

According a result of Mourtada (see [9], Theorem 1), in the vicinity of the parameter value $\lambda=0$, there exists an appropriate domain on the transversal where the fixed-point equation $\pi(x)=x$ has an expansion of the form (1), up to some error term which is innocuous under generic conditions. Here the parameters $(a,r)$ depend smoothly on $\lambda$ and measure respectively the distance between the saddle separatrices and the ratio of eigenvalues at the saddle points. The additional parameter $b>0$ is related to the first multiplier of the Poincaré map along the unperturbed polycycle and is only relevant if the product $r_{1}\cdots r_{n}$ equals one.

In such particular setting, one is interested in finding an upper bound for the cyclicity of the origin. Namely, the number of solutions (1) bifurcating from $x=0$ under small perturbation of the parameters.

Another situation where (1) appears is related to slow-fast systems. More specifically, it is shown in [4] that such equation appears in the study of limit cycles under bifurcation from multi-layer canard cycles. In contrast to the previous situation, here one is interested in finding a global uniform bound for the number of isolated solutions.

Independently, the structure of the group $G$ of real functions generated by translations $x\mapsto x+a$ and power maps $x\mapsto x^{r}$ is a subject of interest in group theory (see e.g. [2]). Here, one is interested in the abstract structure of such group and one natural question is whether $G$ is isomorphic to a free product. From a dynamical systems point of view, this problem is related to the following question, usually called center problem: Characterise those hyperbolic polycycles which lie on the boundary of a continuum of periodic orbits, i.e. those polycycles for which the Poincaré first return map $\pi$ is the identity. We refer the reader to [11] for some results on this direction.

In this paper, we will be interested in estimating the maximum number of real solutions of (1). To state the problem precisely, we need some definitions: Given $(a,r)\in\mathbb{R}^{2n}$, let $I_{n}(a,r)\subset\mathbb{R}$ denote the subset of all real numbers $x$ such that the functions $g_{0},\ldots,g_{n}$ defined by

$$g_{0}(x)=x\quad\mbox{ and }\quad g_{i+1}(x)=a_{i+1}+g_{i}(x)^{r_{i+1}},\mbox{ for }i=1,\ldots,n-1.$$

(2)

are all strictly positive. Since all these functions are strictly monotone (increasing or decreasing), it is easy to see that $I_{n}(a,r)$ is an open (possibly empty) interval of $\mathbb{R}$. If the $r_{i}$ are all positive, $I_{n}(a,r)$ is a nonempty neighborhood of infinity.

So, our goal is to estimate the number of connected components of the solution set

$$S_{n}(b,a,r)=\{x\in I_{n}(a,r)\mid a_{n}+(a_{n-1}+\cdots(a_{1}+x^{r_{1}})^{r_{2}}\cdots)^{r_{n}}-bx=0\},$$

(3)

Let $\mathrm{FP}_{n}(b,a,r)$ denote the number of such connected components. It follows immediately from the analyticity of the functions in (2) that for each fixed parameter $(b,a,r)$, either $S_{n}(b,a,r)\equiv I_{n}(a,r)$ or $S_{n}(b,a,r)$ is a finite set of points. As a consequence, $\mathrm{FP}_{n}(b,a,r)<\infty$, and we define

$$\mathrm{fp}(n)=\sup_{(b,a,r)}\mathrm{FP}_{n}(b,a,r)$$

(4)

The first obvious question is whether $\mathrm{fp}(n)<\infty$. This is a direct Corollary of Khovanskii’s Fewnomial theory [7]. Indeed, by introducing auxiliary variables $(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}_{+}$, equation (1) can be equivalently written as a system,

	$\displaystyle a_{1}+x_{1}^{r_{1}}=x_{2}\hfill$
	$\displaystyle a_{2}+x_{2}^{r_{2}}=x_{3}\hfill$
	$\displaystyle\vdots$
	$\displaystyle a_{n}+x_{n}^{r_{n}}=b\,x_{1}\hfill$

In such form, we have a system of $n$ equations in $n$-variables with $2n+1$ distinct monomials, and the theory of Khovanskii provides the bound

$$\mathrm{fp}(n)\leq\mathrm{K}(n):=2^{n(2n-1)}(n+1)^{2n}$$

(see [7]). In [1], Sotille and Bihan slightly improved the fewnomial’s bound. Their result implies that $\mathrm{fp}(n)\leq\mathrm{SB}(n):=\frac{e^{2}+3}{4}\,2^{\binom{n}{2}}n^{n}$.

One of the difficulties of the problem is that, in general, the sequence (2) is not a Chebyshev system (see e.g. [3], Section 3.3, for the definition). To the author’s knowledge, this was first observed in Mourtada’s thesis [6]. In that work, Mourtada constructed a specific example of a generic hyperbolic polycycle with four singularities which bifurcates into five limit cycles. In our notation, this result translates to $\mathrm{fp}(4)\geq 5$. We shall see in Section 2 that $\mathrm{fp}(4)\leq 13$ and, in Section 4, that in fact $\mathrm{fp}(3)=5$. The exact value of $\mathrm{fp}(n)$ is unknown for $n\geq 4$.

In the next Sections, we shall describe a simple algorithm which gives a new upper bound for $\mathrm{fp}(n)$ by a derivation-division algorithm. The algorithm is somewhat inspired in [8].

Such new upper bound coincides with the exact value of $\mathrm{fp}(n)$ for $n\in\{1,2,3\}$ and is significantly smaller the Khovanskii’s and Bihan-Sottile’s bound for $n\leq 5$. However, it is largely outperformed by such bounds for $n\geq 6$. We refer to Section 6 for a more detailed comparison.

One of the advantages of our method is that it provides as a by-product a system of linearly independent monomials which allows to expand the left-hand side of (1) uniformly with respect to the parameters $(a,r)$. This expansion can be interpreted as a generalization of the compensator-type expansion introduced independently by Roussarie [12] and Ecalle [5]. We refer to Section 5 for the details.

We initially consider a more abstract setting of a certain ring equipped with a derivation.

Given $n\in\mathbb{N}$, let $R=\mathbb{Z}[R_{1},..,R_{n}]$ denote the commutative ring of polynomials $n$ variables $R_{1},\ldots,R_{n}$ with integer coefficients. Our basic object will be the ring

$$\mathbb{L}_{n}=R[x_{1}^{\pm},y_{1}^{\pm},\ldots,x_{n}^{\pm},y_{n}^{\pm}],$$

i.e. the ring of Laurent polynomials in $x_{i},y_{i}$ with coefficients in $R$. We will frequently use the fraction notation and write, for instance, $x_{1}y_{1}^{-1}$ simply as $x_{1}/y_{1}$. We convention that $\mathbb{L}_{0}=R$.

A monomial in $\mathbb{L}_{n}$ is polynomial of the form $\mathfrak{m}=c\,x^{k}y^{l}$, with a nonzero coefficient $c\in R\setminus\{0\}$ and where we note

$$x^{k}y^{l}:=x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}y_{1}^{l_{1}}\cdots y_{n}^{l_{n}}$$

The vector $(k,l)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}$ will be called the exponent $\mathfrak{m}$ and we say $\mathfrak{m}$ unitary if $c=1$. The (poly)-degree of a such monomial is the integer vector $\mathrm{pdeg}(\mathfrak{m})=k+l\in\mathbb{Z}^{n}$. We note that all unitary monomials are invertible element of $\mathbb{L}_{n}$.

Each non-zero polynomial $p\in\mathbb{L}_{n}$ can be written as a finite sum of monomials

$$p=\sum_{i\in I}\mathfrak{m}_{i}$$

(5)

where the set $\mathrm{supp}(p)=\{\mathfrak{m}_{i}:i\in I\}$ will be called the support of $p$. We now define successively three subrings of $\mathbb{L}_{n}$.

$$\mathbb{L}_{n}^{\mathrm{reg}}\subset\mathbb{L}_{n-1}^{\mathrm{hom}}[x_{n},y_{n}]\subset\mathbb{L}_{n}^{\mathrm{hom}}\subset\mathbb{L}_{n}$$

(6)

Firstly, we denote by $\mathbb{L}_{n}^{\mathrm{hom}}\subset\mathbb{L}_{n}$ the subring of Laurent polynomials which are homogeneous, i.e. such that all monomials in their support have a same degree. We will note by $\mathrm{pdeg}(p)\in\mathbb{Z}^{n}$ the degree of an element $p$ of $\mathbb{L}_{n}^{\mathrm{hom}}$. For each $1\leq j\leq n$, we will denote by $\mathrm{pdeg}_{j}(p)\in\mathbb{Z}$ the degree of $p$ with respect to the variables $(x_{j},y_{j})$ (i.e. the $j^{th}$-component of $\mathrm{pdeg}(p)$).

We now define $\mathbb{L}_{n-1}^{\mathrm{hom}}[x_{n},y_{n}]\subset\mathbb{L}_{n}^{\mathrm{hom}}$ as the subring of homogeneous Laurent polynomials which are polynomials in the $(x_{n},y_{n})$-variables. In other words, we consider the Laurent polynomials $p$ such that each monomial $\mathfrak{m}=cx^{k}y^{l}\in\mathrm{supp}(p)$ has an exponent $(k,l)$ satisfying $k_{n},l_{n}\geq 0$. Thus, if we let $m=\mathrm{pdeg}_{n}(p)$ and denote the variables $(x_{n},y_{n})$ simply as $(x,y)$, we can expand such element $p$ in the form

$$p=\sum_{j=0}^{m}q_{j}x^{j}y^{m-j}$$

(7)

with coefficients $q_{j}$ in $\mathbb{L}_{n-1}^{\mathrm{hom}}$.

Finally, we define the smallest subring $\mathbb{L}_{n}^{\mathrm{reg}}\subset\mathbb{L}_{n-1}^{\mathrm{hom}}[x_{n},y_{n}]$ appearing in (6), whose elements will be called $n$-regular polynomials. The definition is by induction on $n$:

• (i)

  For $n=0$, the 0-regular polynomials are simply $L_{0}^{\mathrm{reg}}=R$.

• (ii)

  For $n\geq 1$, we say that a Laurent polynomial $p\in\mathbb{L}_{n-1}^{\mathrm{hom}}[x_{n},y_{n}]$ is $n$-regular if the coefficient $q_{0}$ in the expansion (7) is $(n-1)$-regular.

Note that, in particular, if $p\in\mathbb{L}_{n}^{\mathrm{reg}}$ then $\mathrm{pdeg}(p)\in\mathbb{N}^{n}$ is a positive vector and if $\mathrm{pdeg}_{n}(p)=0$ then $p$ is $(n-1)$-regular. Inductively, we show that $\mathrm{pdeg}(p)=0$ if and only if $p$ belongs to the base ring $R$.

Let us give a different characterisation of $\mathbb{L}_{n}^{\mathrm{reg}}$. Consider a homogeneous Laurent polynomial $p\in\mathbb{L}_{n}^{\mathrm{hom}}$ of degree $\mathrm{pdeg}(p)=(m_{1},..,m_{n})\in\mathbb{N}^{n}$. Then $p$ is $n$-regular if and only if there exists a sequence of nonzero Laurent polynomials $p_{j}\in\mathbb{L}_{j-1}^{\mathrm{hom}}[x_{j},y_{j}]$ for $0\leq j\leq n$ such that:

• (1)

  $p_{n}=p$ and

• (2)

  for each $1\leq j\leq n$, $\mathrm{pdeg}_{j}(p_{j})=m_{j}$ and we can inductively write

  $$p_{j}=Q_{j}+p_{j-1}y_{j}^{m_{j}}$$

  where $Q_{j}\in\mathbb{L}_{j-1}^{\mathrm{hom}}[x_{j},y_{j}]$ is divisible by $x_{j}$ (as a polynomial in $(x_{j},y_{j})$). In other words, $Q_{j}$ contains only monomials of the form $\star x_{j}^{\alpha}y_{j}^{\beta}$ with positive exponents $\alpha,\beta$ such that $\alpha+\beta=m_{j}$ and $\beta<m_{j}$.

This result leads to the following nested form for $p$,

$$p=Q_{n}+(\cdots(Q_{2}+(Q_{1}+\alpha y_{1}^{m_{1}})y_{2}^{m_{2}})\cdots)y_{n}^{m_{n}}$$

(8)

for some $\alpha\in R$.

We now turn $\mathbb{L}_{n}$ into a differential ring by introducing a derivation

$$\partial:\mathbb{L}_{n}\rightarrow\mathbb{L}_{n}$$

which satisfies $\partial(R)=0$ and acts on the variables $x_{j},y_{j}$ according to the following formulas

$$\partial(y_{1})=\partial(x_{1})=\Delta_{1}x_{1},\qquad\partial(y_{j})=\partial(x_{j})=\Delta_{j}x_{j}\,\frac{x_{1}\cdots x_{j-1}}{y_{1}\cdots y_{j-1}},\qquad j\geq 2$$

(9)

where we define $\Delta_{j}=R_{1}\cdots R_{j}\in R$. The following result is an obvious consequence of the above expressions for $\partial$ and the Leibniz rule.

We observe however that the ring $\mathbb{L}_{n}^{\mathrm{reg}}$ is not preserved by $\partial$, as the following simple example shows:

In order to deal with such phenomena, we observe that any element of $\mathbb{L}^{\mathrm{hom}}_{n}$ can be transformed into a regular Laurent polynomial upon division by an appropriate monomial.

More precisely, the regularization monomial associated to a non-zero Laurent polynomial $p\in\mathbb{L}^{\mathrm{hom}}_{n}$ is a unitary monomial $\mathfrak{m}=x^{k}y^{l}$ inductively defined as follows:

• (i)

  Case $n=0$: We define $\mathfrak{m}=1$.

• (ii)

  Case $n\geq 1$: If we note $m=\mathrm{pdeg}_{n}(p)$ then we observe that we can write an expansion of $p$ in the variables $(x,y)=(x_{n},y_{n})$ as

  $$p=\sum_{j=n_{0}}^{n_{1}}q_{j}x^{j}y^{m-j}$$

  for some (possibly negative) integers $n_{0}\leq n_{1}$ and coefficients $q_{j}\in\mathbb{L}_{n-1}^{\mathrm{hom}}$ such that $q_{n_{0}},q_{n_{1}}$ are non-zero. We then define $\mathfrak{m}=x^{n_{0}}y^{d-n_{1}}\bar{\mathfrak{m}}$, where $\bar{\mathfrak{m}}$ is the regularization monomial of $q_{n_{0}}$.

We will denote such monomial $\mathrm{reg}(p)=\mathfrak{m}$. The following result is obvious:

Applying successively the above Theorem, we obtain the following:

In the above setting, we define the derivation-division complexity of $p$ as the smallest integer $m=\mathrm{DD}(p)\geq 0$ such that $p^{(m)}\in R$.

We are now ready to estimate the the real solutions of equation (1).

The following result relates this latter ring with the ring of Laurent polynomials defined in the previous Section.

Consider now the analytic function $\varphi:I(a,r)\rightarrow\mathbb{R}$ given by

$$\varphi(x)=a_{n}+(a_{n-1}+\cdots(a_{1}+x^{r_{1}})^{r_{2}}\cdots)^{r_{n}}-bx$$

(13)

which we can also write as $\varphi=g_{n}-b\,g_{0}$.

An easy computation shows that $\psi=x^{2}\left(\frac{d}{dx}\right)^{2}\varphi$ belongs to $G_{n}$. More specifically, if we consider its image $q=\Phi(\psi)$ under the morphism $\Phi$ given above and the monomial

$$\mathfrak{m}=\frac{x_{1}\cdots x_{n}}{(y_{1}\cdots y_{n-1})^{2}}$$

Then, $p=q/m$ is assumes the very simple form

$$p=\Delta_{n}\sum_{j=1}^{n}(\Delta_{j}-\Delta_{j-1})\;\left(\prod_{i_{1}=1}^{j-1}x_{i_{1}}\right)\left(\prod_{i_{2}=j}^{n-1}y_{i_{2}}\right)$$

(14)

where we define $\Delta_{0}=1$ and recall that $\Delta_{i}=R_{1}\cdots R_{i}$ for $1\leq i\leq n$.

Since the number $\mathrm{DD}(n)$ does not depend on any specific choice of parameters $(a,r)$, we conclude that:

As a simple illustration of the algorithm, let us prove that the equation

$$a_{2}+(a_{1}+x^{r_{1}})^{r_{2}}-b\,x=0$$

(15)

has a maximum of three isolated solutions. If we denote by $\varphi$ the right-hand side of the above equation, the function $\psi=x^{2}\left(\frac{d}{dx}\right)^{2}\varphi$ has the form

$$\psi=r_{1}r_{2}\left(r_{1}\left({r_{2}}-1\right)(a_{1}+x^{r_{1}})^{r_{2}-2}x^{2r_{1}-2}+(r_{1}-1)(a_{1}+x^{r_{1}})^{r_{2}-1}x^{r_{1}-2}\right)$$

Therefore, upon division by $x^{r_{1}}(a+x^{r_{1}})^{r_{2}-2}$ we obtain

$$r_{1}r_{2}\left(r_{1}(r_{2}-1)x^{r_{1}}+(r_{1}-1)(a_{1}+x^{r_{1}})\right)$$

which, under the morphism $\Phi$, gives the regular Laurent polynomial

$$p=A_{1}x_{1}+B_{1}y_{1}$$

where we set $A_{1}=\Delta_{2}(\Delta_{2}-\Delta_{1})$ and $B_{1}=\Delta_{2}(\Delta_{1}-\Delta_{0})$. Now, applying one step of the derivation-division, we obtain

$$A_{1}x_{1}+B_{1}y_{1}\overset{\partial}{\Longrightarrow}\Delta_{1}(A_{1}+B_{1})x_{1}\overset{\cdot/x_{1}}{\Longrightarrow}\Delta_{1}(A_{1}+B_{1})\in R$$

Therefore, $\mathrm{DD}(n)=1$ and we conclude that $\mathrm{fp}(2)\leq 3$.

The following explicit example shows that indeed $\mathrm{fp}(2)=3$. Take $a_{1}=0.004259259259$, $a_{2}=-0.1516666667$, $r_{1}=2$ and $r_{2}=1/3$. Numerical computations with SageMath shows that the equation

$$-0.1516666667+(0.004259259259+x^{2})^{1/3}-x=0$$

has the three isolated solutions $x_{1}=0.0123409...$, $x_{2}=0.1741525...$ and $x_{3}=0.3585065...$.

Let us prove that the equation

$$a_{3}+(a_{2}+(a_{1}+x^{r_{1}})^{r_{2}})^{r_{3}}-b\,x=0$$

(16)

has a maximum of five solutions, for all parameter values $(b,(a,r))\in\mathbb{R}_{>0}\times\mathbb{R}^{6}$.

A similar computation to the previous example leads to the polynomial

$$p=A_{2}x_{1}x_{2}+B_{2}x_{1}y_{2}+C_{2}y_{1}y_{2}$$

where $A_{2}=\Delta_{3}(\Delta_{3}-\Delta_{2})$, $B_{2}=\Delta_{3}(\Delta_{2}-\Delta_{1})$ and $C_{2}=\Delta_{3}(\Delta_{1}-\Delta_{0})$.

The following lines show the derivation-division algorithm ends in at most three steps. We omit the dependence on the coefficients $\Delta_{i}$ to simplify the notation:

$$p^{(0)}=x_{1}x_{2}+x_{1}y_{2}+y_{1}y_{2}$$

$$\partial p^{(0)}=\left(x_{1}+\frac{x_{1}^{2}}{y_{1}}\right)x_{2}+x_{1}y_{2}\Longrightarrow p^{(1)}=\frac{\partial p^{(0)}}{x_{1}}=\left(1+\frac{x_{1}}{y_{1}}\right)x_{2}+y_{2}$$

$$\partial p^{(1)}=\left(\left(\frac{x_{1}}{y_{1}}\right)^{2}+\frac{x_{1}}{y_{1}}\right)x_{2}\Longrightarrow p^{(2)}=\frac{\partial p^{(1)}}{(x_{1}/y_{1}^{2})x_{2}}=x_{1}+y_{1}$$

$$\partial p^{(2)}=x_{1}\Longrightarrow p^{(3)}=\frac{\partial p^{(2)}}{x_{1}}=1$$

Therefore, we conclude that $\mathrm{fp}(3)\leq 5$.