Natural Parameter Conditions for Singular Perturbations of Chemical and Biochemical Reaction Networks

Abstract

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving “small parameters” (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis–Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

Appendix

In this section, we collect some technical matters and proofs, as well as recalling some known results for which a concise presentation seems appropriate and useful.

1.1 Lyapunov Function Arguments

Lyapunov functions can be used to estimate the approach to the slow manifold in a singularly perturbed system, as was mentioned in Introduction. This estimate gives rise to a small parameter $\varepsilon _L$ which controls the distance of the solution to the slow manifold. We give an account of the relevant facts here.

We first state an auxiliary result that goes back to Lyapunov.

Lemma 5

Let Q be a real $n\times n$-matrix, with eigenvalues $\mu _1,\ldots ,\mu _n$, and let $\delta >0$. Then there exists a scalar product $\left\langle \cdot ,\cdot \right\rangle$ on $\mathbb R^n$ such that for all x one has

$$\begin{aligned} \left( \min _{1\le i\le n}\textrm{Re}\,\mu _i -\delta \right) \,\left\langle x,\,x\right\rangle \le \left\langle x,\,Qx\right\rangle \le \left( \max _{1\le i\le n}\textrm{Re}\,\mu _i +\delta \right) \,\left\langle x,\,x\right\rangle , \end{aligned}$$

and

$$\begin{aligned} \left( \min _{1\le i\le n}|\mu _i|^2 -\delta \right) \,\left\langle x,\,x\right\rangle \le \left\langle Qx,\,Qx\right\rangle \le \left( \max _{1\le i\le n}|\mu _i|^2 +\delta \right) \,\left\langle x,\,x\right\rangle . \end{aligned}$$

This can be proven as in Walter (1998, Chapter VII, §30) [see, also Arnold (1992, Chapter 22)]. For matrices that are diagonalizable over $\mathbb C$ build a real basis from real and imaginary parts of a complex eigenbasis. For the non-diagonalizable case, by suitable choice of basis elements the nilpotent part can be chosen to have norm $<\delta$.

1.1.1 Estimates

This presentation follows (Berglund and Gentz 2006, Section 2.1 ff.), but for illustrative purposes, we are satisfied with a local version. Consider a smooth system

$$\begin{aligned} \begin{array}{rcl} \dot{x}&{}= &{}\varepsilon \widetilde{f}_1(x,y,\varepsilon ) \\ \dot{y}&{}= &{} f_2(x,y,\varepsilon ) \end{array},\quad \text {briefly} \begin{pmatrix} \dot{x}\\ \dot{y}\end{pmatrix}= F(\begin{pmatrix} x\\ y\end{pmatrix}) \end{aligned}$$

(89)

with $\begin{pmatrix} x\\ y \end{pmatrix}$ in some open subset of $\mathbb {R}^n$, $x\in \mathbb {R}^m$, and a nonnegative parameter $\varepsilon$. Moreover let $\begin{pmatrix} x_0\\ y_0 \end{pmatrix}$ be such that $f_2(x_0,y_0,0)=0$, and M a suitable compact neighborhood of this point. (More conditions on M will be implicitly imposed below, by further assumptions.)

• Assume furthermore that

  $$\begin{aligned} f_2(x,y,\varepsilon )=0 \Longleftrightarrow y=g(x,\varepsilon ) \end{aligned}$$

  for $\begin{pmatrix} x\\ y \end{pmatrix} \in M$ and $\varepsilon \le \varepsilon _\textrm{max}$, with some positive $\varepsilon _\textrm{max}$ and a smooth function g. The zero set $Y_\varepsilon$ of $f_2(\cdot ,\cdot ,\varepsilon )$ in M will be called the slow manifold, or QSS manifold,²² for $\varepsilon$. By Hadamard’s lemma, after possibly shrinking M there exists a smooth matrix valued function A such that

  $$\begin{aligned} f_2(x,y,\varepsilon )=A(x,y,\varepsilon )\cdot (y-g(x,\varepsilon )). \end{aligned}$$

  Thus, we may rewrite system (89) as

  $$\begin{aligned} \begin{array}{rcl} \dot{x}&{}= &{}\varepsilon \widetilde{f}_1(x,y,\varepsilon ) \\ \dot{y}&{}=&{} A(x,y,\varepsilon )\cdot (y-g(x,\varepsilon )). \end{array} \end{aligned}$$

  (90)

  With

  $$\begin{aligned} D_yf_2(x,y,\varepsilon )=A(x,y,\varepsilon ) + \left( D_yA(x,y,\varepsilon )\,\right) (y-g(x,\varepsilon )), \end{aligned}$$

  one finds in particular

  $$\begin{aligned} D_yf_2(x,y,\varepsilon )=A(x,y,\varepsilon ) \text { on } Y_\varepsilon . \end{aligned}$$

• Now assume that all eigenvalues of $A(x_0,y_0,0)$ have negative real parts. By continuity and suitable choice of M and $\varepsilon _\textrm{max}$, all eigenvalues of $A(x,y,\varepsilon )$ have negative real part for $\begin{pmatrix} x\\ y \end{pmatrix}\in M$ and $0\le \varepsilon \le \varepsilon _\textrm{max}$. Due to Lemma 5, there exists a scalar product $\left\langle \cdot ,\cdot \right\rangle$ on $\mathbb {R}^{n-m}$ and some $\gamma >0$ such that

  $$\begin{aligned} \left\langle z,A(x_0,y_0,0)z\right\rangle \le -2\gamma \left\langle z,z\right\rangle ,\quad \text { all } z\in \mathbb R^{n-m}. \end{aligned}$$

  (Recall the correspondence between $2\gamma$ and eigenvalues.) Thus, we may assume that

  $$\begin{aligned} \left\langle z,A(x,y,\varepsilon )z\right\rangle \le -\gamma \left\langle z,z\right\rangle , \quad \text { all } z\in \mathbb R^{n-m}, \end{aligned}$$

  (91)

  on M, with $0\le \varepsilon \le \varepsilon _\textrm{max}$. Denote by $\Vert \cdot \Vert$ the norm associated with this scalar product.

The following line of arguments is a slight variant of classical reasoning [which uses Gronwall’s lemma, see, e.g., Evans (2014, Appendix B) for the latter]. For solutions of (89) we find

$$\begin{aligned} \begin{array}{rcl} \dfrac{\textrm{d}}{\textrm{d}t} \left\langle y-g(x,\varepsilon ), y-g(x,\varepsilon )\right\rangle &{}=&{}2\left\langle y-g(x,\varepsilon ),\dot{y} -D_xg(x,\varepsilon )f_1(x,y,\varepsilon )\right\rangle \\ &{} =&{} 2\left\langle y-g(x,\varepsilon ),A(x,y,\varepsilon )\left( y-g(x,\varepsilon )\right) \right\rangle \\ &{} &{} \quad -2\left\langle y-g(x,\varepsilon ), D_xg(x,\varepsilon )f_1(x,y,\varepsilon )\right\rangle . \end{array} \end{aligned}$$

The first term on the right-hand side can be estimated by $-\gamma \cdot \langle y-g(x,\varepsilon ), y-g(x,\varepsilon )\rangle$. As for the second term, by Cauchy-Schwarz one has

$$\begin{aligned} \begin{array}{rcl} 2\left| \left\langle y-g(x,\varepsilon ), D_xg(x,\varepsilon )f_1(x,y,\varepsilon )\right\rangle \right| &{} \le &{}2\Vert y-g(x,\varepsilon ) \Vert _2\cdot \Vert D_xg(x,\varepsilon )f_1(x,y,\varepsilon )\Vert _2 \\ &{} \le &{}2\Vert y-g(x,\varepsilon ) \Vert _2\cdot \left( \Vert D_xg(x,\varepsilon )\Vert \cdot \Vert f_1(x,y,\varepsilon )\Vert \right) \end{array} \end{aligned}$$

with suitable norms in the second and third factor.

Now, there exists a positive constant $\kappa =\varepsilon \widetilde{\kappa }$ such that

$$\begin{aligned} \Vert D_xg(x,\varepsilon )\Vert \cdot \Vert f_1(x,y,\varepsilon )\Vert \le \kappa . \end{aligned}$$

So, for $V:=\Vert y-g(x,\varepsilon )\Vert ^2$ one obtains the differential inequality

$$\begin{aligned} \frac{\textrm{d}V}{\textrm{d}t}\le -2\gamma V+2 \kappa \sqrt{V}. \end{aligned}$$

Comparison with the solution of the corresponding Bernoulli equation yields

$$\begin{aligned} V\le V(0)\exp (-\gamma t)+\left( \frac{\kappa }{\gamma }\right) ^2\cdot (1-\exp (-\gamma t)), \end{aligned}$$

(92)

thus $\Vert y-g(x,\varepsilon )\Vert =\sqrt{V(t)}$ can be estimated, e.g., by $\sqrt{2} \frac{\kappa }{\gamma }$ as $t\rightarrow \infty$.²³ Therefore, after a transient phase the proximity of the solution to the slow manifold is controlled by

$$\begin{aligned} \varepsilon _L:=\sqrt{2} \frac{\kappa }{\gamma }=\sqrt{2}\varepsilon \frac{\widetilde{\kappa }}{\gamma }. \end{aligned}$$

(93)

More precisely, once $V(0)\exp (-\gamma t)\le \left( \dfrac{\kappa }{\gamma }\right) ^2$, the stated estimate holds. The inequality is satisfied whenever

$$\begin{aligned} t \ge \frac{1}{\gamma }\,\log \left( \dfrac{\gamma ^2 V(0)}{\kappa ^2} \right) \sim \log \dfrac{1}{\varepsilon _L}, \end{aligned}$$

and this indicates that the time span for the approach to the QSS manifold is of order $|\log \varepsilon _L|$ in the fast timescale, and of order $\varepsilon _L|\log \varepsilon _L|$ in the slow timescale $\varepsilon _L t.$ (A more detailed analysis will provide a lower estimate by a variant of (91), and confirm that the asymptotic estimate cannot be improved.) In particular time spans of order 1 will not suffice for the transient.

In reaction network settings, $\varepsilon _L$ is a dimensional parameter (with dimension concentration); a suitable normalization needs to be chosen.

1.1.2 A Correspondence to Eigenvalues

We sketch the relation of the small parameter $\varepsilon _L$ to eigenvalues of the Jacobian. For the sake of simplicity, we only consider the linearization here, disregarding higher-order terms. Given the system

$$\begin{aligned} \begin{array}{rcccccc} \dot{x} &{} =&{} -\varepsilon \widetilde{U} x&{}+&{}\varepsilon \widetilde{V} y &{} &{} \\ \dot{y} &{} =&{} Wx&{}-&{}Zy&{}=&{} -Z\left( y-Z^{-1}W\right) x\\ \end{array}\quad , \text { briefly} \begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix} =F(x,y,\varepsilon ), \end{aligned}$$

and keeping the notation from above, we have $A=-Z$, $g(x)=Z^{-1}Wx$, $D_xg=Z^{-1}W$. The slow manifold $Y_\varepsilon$ is given by $Wx-Zy=0$, up to higher-order terms. Moreover

$$\begin{aligned} \widetilde{f}_1=-\widetilde{U}x+\widetilde{V} y=\left( -\widetilde{U}+\widetilde{V}Z^{-1}W\right) x \quad \text { on }Y_\varepsilon . \end{aligned}$$

Now consider the eigenvalues of the matrix $DF=\begin{pmatrix}-\varepsilon \widetilde{U}&{}\varepsilon \widetilde{V}\\ W &{} -Z \end{pmatrix}$; see also Lemma 7. Thus, let $\alpha _0+\varepsilon \alpha _1+\cdots$ be an eigenvalue with eigenvector $\begin{pmatrix} x_0+\varepsilon x_1+\cdots \\ y_0+\varepsilon y_1+\cdots \end{pmatrix}$; $\begin{pmatrix} x_0\\ y_0\end{pmatrix}\not = 0$. For $\alpha _0\not =0$, comparing lowest order terms in the eigenvalue condition yields

$$\begin{aligned} x_0=0\text { and } Wx_0-Zy_0=\alpha _0y_0, \end{aligned}$$

thus $-\alpha _0$ is an eigenvalue of Z. By Lemma 5, we see that $2\gamma$ can be chosen near the nonzero eigenvalue of DF(x, y, 0) with smallest absolute real part.

For $\alpha _0=0$, thus the eigenvalue has order $\varepsilon$, comparing lowest orders in the eigenvalue condition yields

$$\begin{aligned} -\widetilde{U} x_0+\widetilde{V} y_0=\alpha _1 x_0\text { and } Wx_0-Zy_0=0, \end{aligned}$$

hence $\alpha _1$ is an eigenvalue for $-\widetilde{U}+\widetilde{V}Z^{-1}W=\widetilde{f}_1$. An upper estimate for $\varepsilon \Vert \widetilde{f}_1\Vert$ can be obtained from Lemma 5: Choose the order $\varepsilon$ eigenvalue with greatest absolute value, multiplied by some factor accounting for a coordinate change. Thus, we see that $\kappa$ is composed of the factor $\Vert Z^{-1}W\Vert$ (which reflects the geometry of the slow manifold), the absolutely largest eigenvalue of order $\varepsilon$ and some multiplicative constants from coordinate transformations. In our local setting, all the multiplicative constants mentioned above are of order one.

To summarize, the small parameter $\varepsilon _L=\kappa /\gamma$ is determined by the ratio of the largest absolute eigenvalue of order $\varepsilon$ to the smallest absolute real part of eigenvalues of order one. From this perspective, for slow manifolds of dimension one in particular, the relevance of the parameters $\varepsilon ^*$ and $\mu ^*$ is obvious. Their advantage lies in their (relative) computational accessibility. Likewise, $\varepsilon ^*$ is a both relevant and computationally accessible parameter for three-dimensional systems with two-dimensional slow manifolds.

1.1.3 Remarks on Steps 2 and 3

Lyapunov function arguments provide a small parameter $\varepsilon _L$ which characterizes closeness of a solution of (89) to the slow manifold. This takes care of Step 1 described in Introduction, and clarifies the role of eigenvalues up to ($\varepsilon$-independent) factors due to coordinate changes.

For the ultimate goal of obtaining quantitative estimates for the discrepancy between the true solution and the singular perturbation approximation, one needs to go further. In Step 2, an appropriate critical time for the onset of the slow dynamics, as well as an appropriate initial value for the reduced system, must be determined. As for Step 3, by a continuity and compactness argument, the right-hand sides of the full and the reduced equation differ by $\varepsilon _L$ times some constant. With this, and an error estimate for the initial value for the reduced system, continuous dependence provides an estimate of the approximation error on compact time intervals. Further work may be required, since one is mostly interested in unbounded time intervals, so one cannot rely only on standard continuous dependence theorems.

In the present manuscript, we generally did not address the determination of $\varepsilon _L$ in examples and case studies. The only exception is irreversible Michaelis–Menten with slow product formation (see, Sect. 4.1), which also contains partial results for Step 3. For the (more familiar and more relevant) irreversible Michaelis–Menten system with small enzyme concentration all three steps can dealt with completely (even if some complications arise), as will be shown in a forthcoming paper. For any system of dimension $>2$, even completing Step 1 seems quite demanding.

1.2 A Proof of Lemma 2

Proof

Part (a) is a special case of Lemma 7 below. To prove part (b), abbreviate $\sigma ^*_i(x):=\sigma _i(x,\widehat{\pi })$ for $x\in \widetilde{Y}\cap K$, $1\le i\le n-1$. Then the nonzero roots of the characteristic polynomial $\chi$ are the roots of

$$\begin{aligned} \zeta (x,\tau ):=\tau ^{n-1}+\sigma ^*_1(x)\tau ^{n-2}+\cdots +\sigma ^*_{n-1}(x). \end{aligned}$$

By the blanket assumptions, the $\sigma ^*_i$ are bounded above and below by positive constants, hence the absolute values of all zeros of the $\zeta (x,\cdot )$ are bounded above by some constant. Since $\widehat{\pi }$ is a TFPV, all zeros have negative real parts. Now assume that for every positive constant $\delta$, some $\zeta (x,\tau )$ has a zero with real part $\ge -\delta$. Then there exist sequences $(x_k)$ in $\widetilde{Y}\cap K$ and $(\mu _k)$ in $\mathbb C$ such that $\zeta (x_k,\mu _k)=0$ and $\textrm{Re}\,\mu _k\rightarrow 0$. Due to boundedness of the sequence $(\mu _k)$ and compactness of $\widetilde{Y}\cap K$ we may assume that the $\mu _k$ converge to $\mu ^*$, $\textrm{Re}\,\mu ^*=0$, and the $x_k$ converge to $x^*\in \widetilde{Y}\cap K$. By continuity $\zeta (x^*,\mu ^*)=0$; a contradiction. Part (c) follows by continuity and compactness arguments.

1.3 Parameter Dependence of Eigenvalues

Recall from (39) the definition

$$\begin{aligned} \widetilde{\sigma }_i(x,\varepsilon ):=\sigma _i(x,\widehat{\pi }+\varepsilon \rho ),\quad 1\le i\le n, \quad \widetilde{\sigma }_0:=1. \end{aligned}$$

We first prove (44), concerning the orders of the $\widetilde{\sigma }_i$ whenever $s>1$.

Lemma 6

Let $\widehat{\pi }$ be a TFPV for dimension s, with critical manifold $\widetilde{Y}$. Then for all $x\in \widetilde{Y}\cap K$ one has

$$\begin{aligned} \widetilde{\sigma }_i(x,\varepsilon )=\varepsilon ^{i-n+s}\widehat{\sigma }_i(x,\varepsilon )\text { for all }x\in \widetilde{Y}\cap K, \quad n-s\le i\le n, \end{aligned}$$

with polynomial $\widehat{\sigma }_i$.

Proof

The arguments we will use are similar to those in the proof of Goeke et al. (2015, Proposition 3). We set

$$\begin{aligned} \widetilde{h}(x,\varepsilon ):=h(x,\widehat{\pi }+\varepsilon \rho )\text { for }x\in \widetilde{Y}\cap K. \end{aligned}$$

There exists a local transformation of $\widetilde{h}$ into Tikhonov standard form. Thus, there exists a local analytic diffeomorphism $\Phi$ and a vector field $\widetilde{q}$ such that

$$\begin{aligned} D\Phi (x)\widetilde{h}(x,\varepsilon )=\widetilde{q}(\Phi (x),\varepsilon ) \end{aligned}$$

and consequently

$$\begin{aligned} D\Phi (x)D\widetilde{h}(x,\varepsilon )=D\widetilde{q}(\Phi (x),\varepsilon )D\Phi (x),\quad x\in \widetilde{Y}\cap K. \end{aligned}$$

Therefore the Jacobian of $\widetilde{h}$ at x and the Jacobian of $\widetilde{q}$ at $\Phi (x)$ are conjugate; in particular they have the same characteristic polynomial. Denoting by $\widetilde{\nu }_i(y)$ the coefficients of the characteristic polynomial of $D\widetilde{q}(y)$, this means

$$\begin{aligned} \widetilde{\sigma }_i(x,\varepsilon )=\widetilde{\nu }_i(\Phi (x),\varepsilon )\text { for all }x\in \widetilde{Y}\cap K. \end{aligned}$$

Since $\widetilde{q}$ is in Tikhonov standard form, we have

$$\begin{aligned} \widetilde{q}(y,\varepsilon )=\begin{pmatrix}\varepsilon \widetilde{q}_1(y,\varepsilon )\\ \widetilde{q}_2(y,\varepsilon )\end{pmatrix}, \end{aligned}$$

with $q_1$ having s entries, and

$$\begin{aligned} D\widetilde{q}(y,\varepsilon )=\begin{pmatrix}\varepsilon D\widetilde{q}_1(y,\varepsilon )\\ D\widetilde{q}_2(y,\varepsilon )\end{pmatrix}. \end{aligned}$$

Thus, every entry of the first s rows of the Jacobian is a multiple of $\varepsilon$, and with the Laplace expansion of the determinant this implies

$$\begin{aligned} \widetilde{\nu }_i(x,\varepsilon )=\varepsilon ^{i-n+s}\,\widehat{\nu }_i(x,\varepsilon ), \quad n-s< i\le n, \end{aligned}$$

and finally (44).

Now we turn to determining the orders of the eigenvalues.

Lemma 7

With objects and notation as in Lemma 6, let (44) hold, and furthermore consider the nondegeneracy conditions:

• (i) $\widehat{\sigma }_{n-s}(x,0)\not =0$ and $\widehat{\sigma }_{n}(x,0)\not =0$ on $\widetilde{Y}\cap K$.

• (ii) The polynomials

  $$\begin{aligned} \widehat{\sigma }_{n-s}(x,0)\tau ^s+ \widehat{\sigma }_{n-s+1}(x,0)\tau ^{s-1}+\cdots +\widehat{\sigma }_{n}(x,0) \end{aligned}$$

  (94)

  admit only simple zeros, for all $x\in \widetilde{Y}\cap K$.

• (a) Whenever (i) holds, the zeros $\lambda _i(x,\varepsilon )$ of the characteristic polynomial can be labeled such that

  $$\begin{aligned} \lambda _1(x,0)\not =0,\ldots , \lambda _{n-s}(x,0)\not =0\quad \text { on } \widetilde{Y}\cap K, \end{aligned}$$

  and

  $$\begin{aligned} \lambda _i(x,\varepsilon )=\varepsilon \widehat{\lambda }_i(x,\varepsilon ),\quad x\in \widetilde{Y}\cap K,\quad i>n-s, \end{aligned}$$

  with continuous $\widehat{\lambda }_i$ such that $\widehat{\lambda }_i(x,0)\not =0$ on $\widetilde{Y}\cap K$, $n-s+1\le i\le n$.²⁴

• (b) Whenever (ii) holds in addition to (i) then all $\widehat{\lambda }_i$, $n-s+1\le i\le n$, are analytic in $(x,\varepsilon )$.

Proof

The proof rests on the Newton–Puiseux theorem and on Hensel’s lemma; we refer specifically to Abhyankar (1990, Lectures 12 and 13). According to Newton–Puiseux, the equation $\lambda ^n+\sum \widetilde{\sigma }_i \lambda ^{n-i}=0$ admits series solutions

$$\begin{aligned} \lambda =\alpha \varepsilon ^\gamma +\cdots \end{aligned}$$

in rational exponents of $\varepsilon$, with a positive rational number $\gamma$ and $\alpha \not =0$. For such an expansion to hold with some $\gamma$ and $\alpha \not =0$, cancellation of lowest order terms in (10) is necessary. The lowest orders of the terms in the monomials are

$$\begin{aligned} (n-i)\gamma \text { for } 0\le i\le s,\quad \text { and } \quad (n-j)\gamma + j-n+s \text { for } s+1\le j\le n, \end{aligned}$$

and for cancellation one must have equality between two of these orders. Clearly two orders in the first block cannot be equal. Assuming that an order from the first block equals an order in the second block, we get

$$\begin{aligned} (n-i)\gamma =(n-j)\gamma + j-n+s\Rightarrow \gamma =\frac{j-(n-s)}{j-i}<1 \text { unless } i=n-s. \end{aligned}$$

But in case $\gamma <1$ the lowest order equals $s\gamma$, with no cancellation; so only $\gamma =1$ remains. Finally, if two orders in the second block are equal then one directly sees $\gamma =1$. This shows part (a).

Continuing the argument, $\gamma =1$ implies that precisely the monomials of degree $\le n-s$ contribute to the lowest order, and the ansatz yields

$$\begin{aligned} \widehat{\sigma }_{n-s}(x,0)\alpha ^s+ \widehat{\sigma }_{n-s+1}(x,0)\alpha ^{s-1}+\cdots +\widehat{\sigma }_{n}(x,0)=0, \end{aligned}$$

thus s distinct choices for $\alpha$ by condition (ii), and $\alpha \not =0$. By Hensel’s lemma, each choice for $\alpha$ yields a series $\lambda =\alpha \varepsilon +\cdots$, in positive integer powers of $\varepsilon$. This shows part (b).

Remark 6

In case $s=1$ the second condition is automatic. Therefore Lemma 2 (a) is also proven.