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Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems

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Abstract

Motivated by the Smale mean value and the Dubinin–Sugawa dual mean value theorems, we formulate non-Archimedean Smale mean value and Dubinin–Sugawa dual mean value problems. We show that if a universal constant satisfying the non-Archimedean Smale mean value problem exists, it must be greater than or equal to one and if a universal constant satisfying the non-Archimedean Dubinin–Sugawa dual mean value problem exists, it must be less than or equal to one. We also derive other bounds whenever roots of unity exist in a non-Archimedean field.

Introduction

In 1981, Steve Smale proved the following result using the Koebe 1/4-theorem [27].

Theorem 1.1. [27] (Smale Mean Value Theorem) Let $P(z) := a_0 + a_1z + a_2z^2 + \cdots+ a_nz^n$ be a polynomial of degree $n \ge2$ over $\mathbb{C}$ . If $z \in\mathbb{C}$ is not a critical point of $P$ (i.e., $P'(z) \ne0$ ), then

\min_{\substack{w \in\mathbb{C},\,P'(w)=0}} \left|\frac{P(z)-P(w)}{z-w}\right| \le4|P'(z)|. \tag*{(1)}

Note that Theorem 1.1 can also be stated as follows.

Theorem 1.2. [27] (Smale Mean Value Theorem) Let $P(z) := a_0 + a_1z + a_2z^2 + \cdots+ a_nz^n$ be a polynomial of degree $n \ge2$ over $\mathbb{C}$ . If $z \in\mathbb{C}$ is not a critical point of $P$ , then there exists a critical point $w \in\mathbb{C}$ of $P$ such that

\left|\frac{P(z)-P(w)}{z-w}\right| \le4|P'(z)|. \tag*{(2)}

The proof of Theorem 1.2 can also be found in books [19, 24]. Inequality (1) led Smale to ask the following problem. What is the best constant $4 > c > 0$ such that the Inequality (1) can be replaced by

\min_{\substack{w \in\mathbb{C},\,P'(w)=0}} \left|\frac{P(z)-P(w)}{z-w}\right| \le c|P'(z)|? \tag*{(3)}

Smale made the following conjecture with regard to Inequality (3), which remains open today.

Conjecture 1.3. [25, 26, 27, 28, 29, 30] (Smale Mean Value Conjecture) Let $P(z):=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq2$ over $\mathbb{C}$ . If $z\in\mathbb{C}$ is not a critical point of $P$ , then

\min_{\substack{w\in\mathbb{C},P'(w)=0}}\left|\frac{P(z)-P(w)}{z-w}\right|\leq|P'(z)|

\min_{\substack{w\in\mathbb{C},P'(w)=0}}\left|\frac{P(z)-P(w)}{z-w}\right|\leq\frac{n-1}{n}|P'(z)|=\frac{\deg(P)-1}{\deg(P)}|P'(z)|. \tag*{(4)}

By taking polynomial $P(z):=z^n-nz\in\mathbb{C}[z]$ , we see that the constant $(n-1)/n$ in inequality (4) cannot be improved further. Conjecture 1.3 also appears as Problem 1 in the Addenda after the list of 18 open problems of Smale in Mathematical Problems for the Next Century [30]. Now for a polynomial $P$ of degree $n\geq2$ , define

S(P):=\sup_{\substack{z\in\mathbb{C},P'(z)\neq0}}\frac{1}{|P'(z)|}\min_{\substack{w\in\mathbb{C},P'(w)=0}}\left|\frac{P(z)-P(w)}{z-w}\right|

and

M:=\sup_{P\text{ is a non linear polynomial}}S(P).

Smale conjecture then asks whether $M=1$ or $S(P)=\frac{\deg(P)-1}{\deg(P)}$ ? It is known that the Conjecture 1.3 holds for polynomials of degree 2, 3, 4, [19, 24, 31]. Through computer calculations, Smale conjecture is known to hold for polynomials of degree upto 10 [23].

Usually, the study of Smale mean value conjecture goes with the following conjecture which is known as normalization equivalence to Conjecture 1.3.

Conjecture 1.4. [19, 24, 27] (Normalized Smale Mean Value Conjecture) Let $P$ be a degree $n\geq2$ polynomial over $\mathbb{C}$ which is normalized, i.e., $P(0)=0$ and $P'(0)=1$ . Then

S_0(P):=\min_{\substack{w\in\mathbb{C},P'(w)=0}}\left|\frac{P(w)}{w}\right|=\frac{\deg(P)-1}{\deg(P)}

M_0:=\sup_{\substack{P\text{ is a non linear polynomial},P(0)=0,P'(0)=1}}S_0(P)=1.

In 1996, Andrievskii and Ruscheweyh showed that if critical points of a normalized polynomial $P$ lie on the unit circle, then $S_0(P)\leq1$ [1]. In 2009, Hinkkanen and Kayumov showed that if all the critical points of a normalized polynomial $P$ are real, then $S_0(P)\leq\frac{2}{3}$ [12]. In 2010, Hinkkanen and Kayumov proved that if the critical points of normalized polynomial $P$ lie on the sector $\{re^{i\theta}:r>0,|\theta|\leq\pi/6\}$ , then $S_0(P)\leq\frac{1}{2}$ and if the critical points of normalized polynomial $P$ lie in the union of two rays $\{1+re^{\pm i\theta}:r\geq0\}$ for each $0<\theta\leq\pi/2$ , then $S_0(P)\leq\frac{2}{3}$ [13]. In [19] it is showed that if the critical points of a normalized polynomial are all real, then $S_0(P)\leq\frac{\deg(P)}{\deg(P)}\left[\left(\frac{\deg(P)-1}{\deg(P)-2}\right)^{\deg(P)-1}-2\right]$ and in [24] it is showed that $S_0(P)\leq e-2$ .

In 2002, Beardon, Minda and Ng showed that $S(P)\leq4^{\frac{\deg(P)-2}{\deg(P)-1}}$ [2]. In 2006, Fijikawa and Sugawa proved that $S(P)\leq4^{\frac{1+(\deg(P)-2)4^{\frac{\deg(P)-1}{\deg(P)+1}}}{\deg(P)+1}}$ [11]. In 2007, Conte, Fujikawa and Lakic showed that $S(P)\leq4^{\frac{\deg(P)-1}{\deg(P)+1}}$ [3]. In 2007, Crane showed that there exists $n_0\geq2$ such that for all $n\geq n_0$ , $S(P)\leq4-\frac{2}{\sqrt{\deg(P)}}$ [5] (also see [4]). In 2003, Ng proved that Conjecture 1.3 holds with $M=2$ for all odd polynomials with nonzero linear terms [16]. In 2006, Dubinin proved that if the critical points of a normalized polynomial $P$ have equal modulus, then $S_0(P) \le\frac{\deg(P)-1}{\deg(P)}$ [10].

At the other side, in 2002, Sendov formulated a generalization of Conjecture 1.3 [21]. In 2003, Sendov and Nikolov gave some variations of Conjecture 1.3 [22]. In 2005, Tyson showed that Tischler’s strong form of Smale mean value conjecture [31, 32] fails [33]. One can interpret Conjecture 1.3 in the form of finite Blaschke products [17, 18, 24]. It is also possible to connect Conjecture 1.3 with electrostatics [6]. In 1981, Smale also derived the following higher order mean value theorem [27].

Theorem 1.5. [27] (Higher Order Smale Mean Value Theorem) Let $P(z) := a_0 + a_1z + a_2z^2 + \cdots+ a_nz^n$ be a polynomial of degree $n \ge2$ over $\mathbb{C}$ . If $z \in\mathbb{C}$ is not a critical point of $P$ , then there exists a critical point $w \in\mathbb{C}$ of $P$ such that

\frac{|P^{(k)}(z)|}{k!}\frac{|P(z)-P(w)|^{k-1}}{|P'(z)|^k} \le4^{k-1}, \qquad\forall2 \le k \le n. \tag*{(5)}

In 2009, Dubinin and Sugawa derived the following striking result which is a dual result for Smale mean value theorem [9].

Theorem 1.6. [7, 9] (Dubinin-Sugawa Dual Mean Value Theorem) Let $P(z) := a_0 + a_1z + a_2z^2 + \cdots+ a_nz^n$ be a polynomial of degree $n \ge2$ over $\mathbb{C}$ . If $z \in\mathbb{C}$ is not a critical point of $P$ , then there exists a critical point $w \in\mathbb{C}$ of $P$ such that

\frac{|P'(z)|}{n4^n} \le\left|\frac{P(z)-P(w)}{z-w}\right|.

In other words,

\frac{|P'(z)|}{n4^n} \le\max_{w\in\mathbb{C},P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|.

Dubinin and Sugawa made the following conjecture in 2009 [9].

Conjecture 1.7. [7, 9] (Dubinin-Sugawa Dual Mean Value Conjecture) Let $P(z) := a_0 + a_1z + a_2z^2 + \cdots+ a_nz^n$ be a polynomial of degree $n \ge2$ over $\mathbb{C}$ . If $z \in\mathbb{C}$ is not a critical point of $P$ , then

\frac{|P'(z)|}{n} \le\max_{w\in\mathbb{C},P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|.

For a polynomial $P$ of degree $n \ge2$ , define

DS(P) := \inf_{z\in\mathbb{C},P'(z)\ne0}\frac{1}{|P'(z)|}\max_{w\in\mathbb{C},P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|.

Similar to Conjecture 1.4, we have the following normalized version of Conjecture 1.7.

Conjecture 1.8. [7, 9] (Normalized Smale Dubinin-Sugawa Dual Mean Value Conjecture) Let $P$ be a degree $n \ge2$ polynomial over $\mathbb{C}$ which is normalized, i.e., $P(0)=0$ and $P'(0)=1$ . Then

DS_0(P) := \max_{w\in\mathbb{C},P'(w)=0}\left|\frac{P(w)}{w}\right| = \frac{1}{n} = \frac{1}{\deg(P)}.

In 2016, Ng and Zhang showed that if $P$ is a normalized polynomial, then $DS_0(P) \ge\frac{1}{4\deg(P)}$ [17]. In 2019 Dubinin showed that $DS(P) \ge\frac{1}{4}\tan\left(\frac{\pi}{4\deg(P)}\right)$ [8]. In 2019, Hinkkanen, Kayumov and Khammatova showed that Conjecture 1.7 holds for polynomials of degree six [14]. In 2025, Hinkkanen, Kayumov and Khammatova showed that Conjecture 1.7 holds for polynomials of degree six [15].

Non-Archimedean Smale Mean Value Problem

Let $\mathbb{K}$ be a field. Recall that a map $|\cdot|:\mathbb{K}\to[0,\infty)$ is said to be a non-Archimedean valuation if the following conditions hold:

(i) If $\lambda\in\mathbb{K}$ satisfies $|\lambda|=0$ , then $\lambda=0$ .

(ii) $|\lambda\mu|=|\lambda||\mu|$ for all $\lambda,\mu\in\mathbb{K}$ .

(iii) (Ultratriangle inequality) $|\lambda+\mu|\le\max\{|\lambda|,|\mu|\}$ for all $\lambda,\mu\in\mathbb{K}$ .

We say $\mathbb{K}$ is non-Archimedean complete valued field if it is complete w.r.t. the ultrametric $d(\lambda,\mu):=|\lambda-\mu|$ for all $\lambda,\mu\in\mathbb{K}$ [20]. Throughout the article we assume that $\mathbb{K}$ is algebraically closed.

We ask for non-Archimedean version of Theorem 1.2 as follows.

Problem 2.1. (Non-Archimedean Smale Mean Value Problem) Let $\mathbb{K}$ be a non-Archimedean (complete) valued field. Whether there is a universal constant $S_{\mathbb{K}}$ (which may depend upon $\mathbb{K}$ ) satisfying the following: Let $P(z):=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\ge2$ over $\mathbb{K}$ . If $z\in\mathbb{K}$ is not a critical point of $P$ , then there exists a critical point $w\in\mathbb{K}$ of $P$ such that

\left|\frac{P(z)-P(w)}{z-w}\right|\le S_{\mathbb{K}}|P'(z)|.

In the following result we derive a constraint for non-Archimedean Smale mean value constant, if exists.

Theorem 2.2. Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $S_{\mathbb{K}}$ in Problem 2.1, if exists, satisfies $S_{\mathbb{K}}\ge1$ .

Proof. Let $P(z):=(z-a)(z-b)\in\mathbb{K}[z]$ . Then $P'(z)=2z-(a+b)$ for all $z\in\mathbb{K}$ . Let $z\in\mathbb{K}$ be such that $P'(z)\ne0$ . Note that $P'$ has only one zero, namely $c:=\frac{a+b}{2}$ . Define $x:=z-a$ and $y:=z-b$ . Then

z-c=z-\frac{a+b}{2}=\frac{z-a+z-b}{2}=\frac{x+y}{2},

P'(z)=z-a+z-b=x+y,

P(c)=\left(\frac{a+b}{2}-a\right)\left(\frac{a+b}{2}-b\right)=-\frac{(b-a)^2}{4}=-\frac{(x-y)^2}{4}.

and

P(z)-P(c)=xy+\frac{(x-y)^2}{4}=\frac{(x+y)^2}{4}.

Hence

\left|\frac{P(z)-P(c)}{z-c}\right|=\left|\frac{x+y}{2}\right|=\frac{1}{|2|}|P'(z)|.

Therefore

S_{\mathbb{K}}\ge\frac{1}{|2|}\ge1.

:::

Theorem 2.3. Let $\mathbb{K}$ be a non-Archimedean valued field. Assume that there is an $n\in\mathbb{N}\setminus\{1,2\}$ such that $\mathbb{K}$ has $(n-1)$ -th roots of unity. The constant $S_{\mathbb{K}}$ in Problem 2.1, if exists, satisfies $S_{\mathbb{K}}\ge\frac{|n-1|}{|n|}=\left|1-\frac{1}{n}\right|$ .

Proof. We define the polynomial $P(z) := z^n - nz \in\mathbb{K}[z]$ . The point $z = 0$ is not a critical point of $P$ . Let $w \in\mathbb{K}$ be any critical point of $P$ . Then $w$ is a $(n-1)$ -th root of unity. We now see that

\begin{aligned} \frac{|P(0)-P(w)|}{|0-w|}=\frac{|P(w)|}{|w|}=|w^n-nw|=|w^{n-1}-n| \\ &=|1-n|=\frac{|1-n|}{|n|}|n|=\frac{|1-n|}{|n|}|P'(0)|. \end{aligned}

Hence $S_{\mathbb{K}} \ge\frac{|n-1|}{|n|}=\left|1-\frac{1}{n}\right|$ .

Based on Theorem 1.5 we formulate the following higher order problem.

Problem 2.4. (Higher Order Non-Archimedean Smale Mean Value Problem) Let $\mathbb{K}$ be a non-Archimedean (complete) valued field. Whether there is a universal constant $R_{\mathbb{K}}$ (which may depend upon $\mathbb{K}$ ) satisfying the following: Let $P(z):=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\ge2$ over $\mathbb{K}$ . If $z\in\mathbb{K}$ is not a critical point of $P$ , then there exists a critical point $w\in\mathbb{K}$ of $P$ such that

\frac{|P^{(k)}(z)|}{k!}\frac{|P(z)-P(w)|^{k-1}}{|P'(z)|^k}\le R_{\mathbb{K}}^{k-1},\qquad\forall2\le k\le n.

Theorem 2.5. Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $R_{\mathbb{K}}$ in Problem 2.4, if exists, satisfies $R_{\mathbb{K}}\ge1/(2|2|)$ .

Proof. We continue from the proof of Theorem 2.2. Let $z\in\mathbb{K}$ be not a zero of $P'$ . We have $P''(z)=2$ . Hence

|P''(z)|=|2|,\qquad|P'(z)|=|x+y|,\qquad|P(z)-P(c)|=\frac{|x+y|^2}{|4|}.

We find

\frac{|P^{(2)}(z)|}{2!}\frac{|P(z)-P(c)|^{2-1}}{|P'(z)|^2} =\frac{|2|}{2}\frac{\left(\frac{|x+y|^2}{|4|}\right)}{|x+y|^2} =\frac{1}{2|2|}.

Therefore

R_{\mathbb{K}}\ge\frac{1}{2|2|}.

Non-Archimedean Dubinin-Sugawa Dual Mean Value Problem

We ask non-Archimedean version of Theorem 1.6 as follows.

Problem 3.1. (Non-Archimedean Dubinin-Sugawa Dual Mean Value Problem) Let $\mathbb{K}$ be a non-Archimedean (complete) valued field. Whether there is a universal constant $D_{\mathbb{K}}$ (which may depend upon $\mathbb{K}$ ) satisfying the following: Let $P(z):=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\ge2$ over $\mathbb{K}$ . If $z\in\mathbb{K}$ is not a critical point of $P$ , then there exists a critical point $w\in\mathbb{K}$ of $P$ such that

D_{\mathbb{K}}|P'(z)|\le\frac{|P(z)-P(w)|}{|z-w|}.

Theorem 3.2. Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $D_{\mathbb{K}}$ in Problem 3.1, if exists, satisfies $D_{\mathbb{K}}\le1$ .

Proof. From the proof of Theorem 2.2, we get that $D_{\mathbb{K}}\le|2|\le1$ .

Theorem 3.3. Let $K$ be a non-Archimedean valued field. Assume that there is an $n \in\mathbb{N} \setminus\{1,2\}$ such that $K$ has $(n-1)$ -th roots of unity. The constant $D_K$ in Problem 3.1, if exists, satisfies $D_K \le\frac{|n|}{|n-1|}$ .

Proof. By defining polynomial $P(z) := z^n - nz \in K[z]$ we get that $D_K \le\frac{|n|}{|n-1|}$ .

Conclusion

For 45 years, the Smale mean value theorem has remained a cornerstone result in the theory of polynomials, while the Dubinin–Sugawa dual mean value theorem has served a similar vital role for 17 years. In this article, we asked for non-Archimedean versions of both theorems. We established explicit bounds for these constants, conditioning on their existence.

References

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Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems

Abstract

Introduction

Non-Archimedean Smale Mean Value Problem

Non-Archimedean Dubinin-Sugawa Dual Mean Value Problem

Conclusion

References

References

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