WorksMathematics

Noncommutative Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda Bounds for Roots of Polynomials

K. MAHESH KRISHNA

Published Oct 31, 2026Fresh — published within the last 30 daysv13 min read

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Abstract

Two centuries ago, Lagrange derived an upper bound for every root of a complex polynomial using the maximum of powers of coefficients. In 1916, Fujiwara independently found the Lagrange bound. In 1914, Carmichael and Mason derived an upper bound using the sum of squares of the absolute values of the coefficients of a polynomial. This bound was extended by Kuniyeda in 1916. We derive noncommutative versions of the Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda bounds.

Introduction

More than 200 years ago, Lagrange derived an upper bound for the zeros of complex polynomials [10, 13]. This bound was later found independently by Fujiwara [5] in 1916.

Theorem 1.1. [5, 6, 10, 13] (Lagrange-Fujiwara Upper Bound) Let

p(z) = a_0 + a_1 z + \cdots+ a_{n-1}z^{n-1} + z^n \in\mathbb{C}[z].

If $\lambda\in\mathbb{C}$ satisfies $p(\lambda) = 0$ , then

|\lambda| < 2 \max_{0 \leq j \leq n-1} |a_j|^{\frac{1}{n-j}}.

Corollary 1.2. [5, 6, 10, 13] (Lagrange-Fujiwara Upper Bound) Let

p(z) = a_0 + a_1 z + \cdots+ a_{n-1}z^{n-1} + a_nz^n \in\mathbb{C}[z], \qquad a_n \ne0.

If $\lambda\in\mathbb{C}$ satisfies $p(\lambda) = 0$ , then

|\lambda| < 2 \max_{0 \leq j \leq n-1} \left|\frac{a_j}{a_n}\right|^{\frac{1}{n-j}}.

Corollary 1.2 gives the following result.

Theorem 1.3. [5, 6, 10, 13] (Lagrange-Fujiwara Lower Bound)

p(z) = a_0 + a_1 z + \cdots+ a_{n-1}z^{n-1} + a_nz^n \in\mathbb{C}[z], \qquad a_0 \ne0, \ a_n \ne0.

If $\lambda\in\mathbb{C}$ satisfies $p(\lambda)=0$ , then

|\lambda|>\frac{1}{2\max_{1\leq j\leq n}\left|\frac{a_j}{a_0}\right|^{\frac{1}{j}}}

In 1914, Carmichael and Mason derived the following beautiful upper bound (using Hadamard theorem for determinants) for every root of a complex polynomial using the sum of the squares of the absolute values of the coefficients [3].

Theorem 1.4. [3, 4, 8, 11] (Carmichael-Mason Upper Bound) Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathbb{C}[z].

If $\lambda\in\mathbb{C}$ satisfies $p(\lambda)=0$ , then

|\lambda|<\left(1+\sum_{j=0}^{n-1}|a_j|^2\right)^{\frac{1}{2}}.

In 1916, Kuniyeda improved Theorem 1.4 using the Holder inequality [7].

Theorem 1.5. [1, 2, 7, 8, 9, 11, 12] (Kuniyeda Upper Bound) Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in\mathbb{C}[z].

If $\lambda\in\mathbb{C}$ satisfies $p(\lambda)=0$ , then

|\lambda|<\left(1+\left(\sum_{j=0}^{n-1}|a_j|^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}},\qquad\forall s\in(1,\infty).

Corollary 1.6. [1, 2, 7, 8, 9, 11, 12] (Kuniyeda Upper Bound) Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in\mathbb{C}[z],\quad a_n\ne0.

If $\lambda\in\mathbb{C}$ satisfies $p(\lambda)=0$ , then

|\lambda|<\left(1+\left(\sum_{j=0}^{n-1}\left|\frac{a_j}{a_n}\right|^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}}=\left(1+\left(\frac{1}{|a_n|^s}\sum_{j=0}^{n-1}|a_j|^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}},\qquad\forall s\in(1,\infty).

Corollary 1.6 gives the following result.

Theorem 1.7. [1, 2, 7, 8, 9, 11, 12] (Kuniyeda Lower Bound) Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in\mathbb{C}[z],\quad a_0\ne0,a_n\ne0.

If $\lambda\in\mathbb{C}$ satisfies $p(\lambda)=0$ , then

|\lambda|>\frac{1}{\left(1+\left(\frac{1}{|a_0|^s}\sum_{j=1}^{n}|a_j|^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}}},\qquad\forall s\in(1,\infty).

We naturally ask for versions of Theorems 1.1 and 1.5 for polynomials over Banach algebras (particularly, over C*-algebras). In this article, we derive them.

Noncommutative Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda Bounds

Let $\mathcal{A}$ be a unital Banach algebra and $\mathcal{A}[z]$ be the set of all polynomials over $\mathcal{A}$ . We derive a noncommutative version of Theorem 1.1 as follows.

Theorem 2.1. (Noncommutative Lagrange-Fujiwara Upper Bound) Let $\mathcal{A}$ be a unital Banach algebra. Let

p(z) = a_0 + a_1 z + \cdots+ a_{n-1}z^{n-1} + z^n \in\mathcal{A}[z].

Let $u \in\mathcal{A}$ be such that $p(u)=0$ and

\lVert u^n\rVert= \lVert u\rVert^n. \tag*{(1)}

Then

\lVert u\rVert< 2 \max_{0\leq j\leq n-1} \lVert a_j\rVert^{\frac{1}{n-j}}.

Proof. Let $u \in\mathcal{A}$ satisfies $p(u)=0$ and $\lVert u^n\rVert= \lVert u\rVert^n$ . Define

M := \max_{0\leq j\leq n-1} \lVert a_j\rVert^{\frac{1}{n-j}}.

Then we have

M^{n-j}(2) \geq\lVert a_j\rVert,\qquad\forall\,0\leq j\leq n-1. \tag*{(2)}

If $\lVert u\rVert\leq M$ , then the conclusion holds. Hence we assume that $\lVert u\rVert> M$ . Since $p(u)=0$ , we have

u^n = -(a_0+a_1u+\cdots+a_{n-1}u^{n-1}).

By taking the norm and using Equation (1), we get

\begin{aligned} \lVert u\rVert^n &= \lVert u^n\rVert = \lVert-(a_0+a_1u+\cdots+a_{n-1}u^{n-1})\rVert\\ &= \lVert a_0+a_1u+\cdots+a_{n-1}u^{n-1}\rVert\\ &\leq\lVert a_0\rVert+\lVert a_1u\rVert+\cdots+\lVert a_{n-1}u^{n-1}\rVert\\ &\leq\lVert a_0\rVert+\lVert a_1\rVert\lVert u\rVert+\cdots+\lVert a_{n-1}\rVert\lVert u^{n-1}\rVert\\ &\leq\lVert a_0\rVert+\lVert a_1\rVert\lVert u\rVert+\cdots+\lVert a_{n-1}\rVert\lVert u\rVert^{n-1}. \end{aligned}

Dividing by $\lVert u\rVert^n$ and by using Equation (2) we get

\begin{aligned} 1 &\le\frac{\lVert a_0\rVert}{\lVert u\rVert^n}+\frac{\lVert a_1\rVert}{\lVert u\rVert^{n-1}}+\cdots+\frac{\lVert a_{n-2}\rVert}{\lVert u\rVert^2}+\frac{\lVert a_{n-1}\rVert}{\lVert u\rVert}\\ &\le\frac{M^n}{\lVert u\rVert^n}+\frac{M^{n-1}}{\lVert u\rVert^{n-1}}+\cdots+\frac{M^2}{\lVert u\rVert^2}+\frac{M}{\lVert u\rVert}\\ &=\frac{M}{\lVert u\rVert}\left(\left(\frac{M}{\lVert u\rVert}\right)^{n-1}+\left(\frac{M}{\lVert u\rVert}\right)^{n-2}+\cdots+\frac{M}{\lVert u\rVert}+1\right)\\ &=\frac{M}{\lVert u\rVert}\left(\frac{1-\left(\frac{M}{\lVert u\rVert}\right)^n}{1-\frac{M}{\lVert u\rVert}}\right)\\ &<\frac{M}{\lVert u\rVert}\left(\frac{1}{1-\frac{M}{\lVert u\rVert}}\right)\\ &=\frac{M}{\lVert u\rVert-M}. \end{aligned}

Hence

1<\frac{M}{\lVert u\rVert-M}.

Rearranging completes the proof. □

Note that we used the condition (1) in the proof of Theorem 2.1. We are unable to derive the result without this assumption.

Corollary 2.2. (Noncommutative Lagrange-Fujiwara Upper Bound) Let $A$ be a unital Banach algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in A[z],\qquad a_n\text{ is invertible}.

Let $u\in A$ be such that $p(u)=0$ and

\lVert u^n\rVert=\lVert u\rVert^n.

Then

\lVert u\rVert<2\max_{0\le j\le n-1}\left\lVert a_n^{-1}a_j\right\rVert^{\frac{1}{n-j}}.

Corollary 2.3. Let $A$ be a unital $C^*$ -algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in A[z],\qquad a_n\text{ is invertible}.

Let $u\in A$ be such that $p(u)=0$ and $u$ is normal or self-adjoint. Then

\lVert u\rVert<2\max_{0\le j\le n-1}\left\lVert a_n^{-1}a_j\right\rVert^{\frac{1}{n-j}}.

Corollary 2.4. (Noncommutative Lagrange-Fujiwara Lower Bound) Let $A$ be a unital Banach algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in A[z],\qquad a_0\text{ and }a_n\text{ are invertible}.

Let $u\in A$ be such that $p(u)=0$ , $u$ is invertible and

\lVert(u^{-1})^n\rVert=\lVert u^{-1}\rVert^n.

Then

\frac{1}{\lVert u^{-1}\rVert}>\frac{1}{2\max_{1\le j\le n}\lVert a_0^{-1}a_j\rVert^{\frac{1}{j}}}.

Proof. Since $u$ is invertible, we have

0=a_n+a_{n-1}u^{-1}+\cdots+a_1u^{-(n-1)}+a_0u^{-n}.

Define $v:=u^{-1}$ . Then

0=a_n+a_{n-1}v+\cdots+a_1v^{n-1}+a_0v^n.

Result follows by applying Corollary 2.2 to $v$ . □

We derive a noncommutative version of Theorem 1.5 as follows.

Theorem 2.5. (Noncommutative Kuniyeda Upper Bound) Let $A$ be a unital Banach algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in A[z].

Let $u\in A$ be such that $p(u)=0$ and

\lVert u^n\rVert=\lVert u\rVert^n. \tag*{(3)}

Then

\lVert u\rVert<\left(1+\left(\sum_{j=0}^{n-1}\lVert a_j\rVert^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}},\qquad\forall s\in(1,\infty).

Proof. Let $s\in(1,\infty)$ . Define

M:=\left(\sum_{j=0}^{n-1}\lVert a_j\rVert^s\right)^{\frac{1}{s-1}}.

Let $u\in A$ satisfies $p(u)=0$ and $\lVert u^n\rVert=\lVert u\rVert^n$ . We claim that

\lVert u\rVert<(1+M)^{\frac{s-1}{s}}.

Let us suppose that the claim fails. Then

\lVert u\rVert\ge(1+M)^{\frac{s-1}{s}}>1.

Since $p(u)=0$ , we have

u^n=-(a_0+a_1u+\cdots+a_{n-1}u^{n-1}).

By taking the norm, using the Holder inequality and using Equation (3), we get

\begin{aligned} \lVert u\rVert^n &= \lVert u^n\rVert= \lVert-(a_0+a_1u+\cdots+a_{n-1}u^{n-1})\rVert\\ &= \lVert a_0+a_1u+\cdots+a_{n-1}u^{n-1}\rVert\\ &\le\lVert a_0\rVert+\lVert a_1u\rVert+\cdots+\lVert a_{n-1}u^{n-1}\rVert\\ &\le\lVert a_0\rVert+\lVert a_1\rVert\lVert u\rVert+\cdots+\lVert a_{n-1}\rVert\lVert u^{n-1}\rVert\\ &\le\lVert a_0\rVert+\lVert a_1\rVert\lVert u\rVert+\cdots+\lVert a_{n-1}\rVert\lVert u\rVert^{n-1}\\ &=\sum_{j=0}^{n-1}\lVert a_j\rVert\lVert u\rVert^j\\ &\le\left(\sum_{j=0}^{n-1}\lVert a_j\rVert^s\right)^{\frac{1}{s}}\left(\sum_{j=0}^{n-1}\left(\lVert u\rVert^j\right)^{\frac{s}{s-1}}\right)^{\frac{s-1}{s}}\\ &=M^{\frac{s-1}{s}}\left(\sum_{j=0}^{n-1}\left(\lVert u\rVert^{\frac{s}{s-1}}\right)^j\right)^{\frac{s-1}{s}}\\ &=M^{\frac{s-1}{s}}\left(\frac{\left(\lVert u\rVert^{\frac{s}{s-1}}\right)^n-1}{\lVert u\rVert^{\frac{s}{s-1}}-1}\right)^{\frac{s-1}{s}}\\ &=M^{\frac{s-1}{s}}\left(\frac{\lVert u\rVert^{\frac{ns}{s-1}}-1}{\lVert u\rVert^{\frac{s}{s-1}}-1}\right)^{\frac{s-1}{s}}. \end{aligned}

Hence

\lVert u\rVert^{\frac{ns}{s-1}}\le M\frac{\lVert u\rVert^{\frac{ns}{s-1}}-1}{\lVert u\rVert^{\frac{s}{s-1}}-1}.

Rearranging,

\lVert u\rVert^{\frac{s}{s-1}}-1\le M\frac{\lVert u\rVert^{\frac{ns}{s-1}}-1}{\lVert u\rVert^{\frac{ns}{s-1}}}=M\left(1-\frac{1}{\lVert u\rVert^{\frac{ns}{s-1}}}\right)<M.

Therefore

(1+M)^{\frac{s-1}{s}}\le\lVert u\rVert<(1+M)^{\frac{s-1}{s}}

which is a contraction. Hence claim holds which completes the proof. □

Note that we used the condition (3) in the proof of Theorem 2.5. We are unable to derive the result without this assumption.

Corollary 2.6. (Noncommutative Kuniyeda Upper Bound) Let $A$ be a unital Banach algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in A[z],\qquad a_n\text{ is invertible}.

Let $u\in A$ be such that $p(u)=0$ and

\lVert u^n\rVert=\lVert u\rVert^n.

Then

\lVert u\rVert< \left(1+\left(\sum_{j=0}^{n-1}\lVert a_n^{-1}a_j\rVert^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}} \le\left(1+\left(\lVert a_n^{-1}\rVert^s\sum_{j=0}^{n-1}\lVert a_j\rVert^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}}, \qquad\forall s\in(1,\infty).

Corollary 2.7. (Noncommutative Kuniyeda Lower Bound) Let $\mathcal A$ be a unital Banach algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in\mathcal A[z],\qquad a_0\text{ and }a_n\text{ are invertible}.

Let $u\in\mathcal A$ be such that $p(u)=0$ , $u$ is invertible and

\lVert(u^{-1})^n\rVert=\lVert u^{-1}\rVert^n.

Then

\frac{1}{\lVert u^{-1}\rVert}>\frac{1}{\left(1+\left(\sum_{j=1}^{n}\lVert a_0^{-1}a_j\rVert^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}}}\ge\frac{1}{\left(1+\left(\lVert a_0^{-1}\rVert^s\sum_{j=1}^{n}\lVert a_j\rVert^s\right)^{\frac{1}{s-1}}\right)^{\frac{s-1}{s}}}.

Corollary 2.8. (Noncommutative Carmichael-Mason Upper Bound) Let $\mathcal A$ be a unital Banach algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in\mathcal A[z],\qquad a_n\text{ is invertible}.

Let $u\in\mathcal A$ be such that $p(u)=0$ and

\lVert u^n\rVert=\lVert u\rVert^n.

Then

\lVert u\rVert<\left(1+\sum_{j=0}^{n-1}\lVert a_n^{-1}a_j\rVert^2\right)^{\frac{1}{2}}\le\left(1+\lVert a_n^{-1}\rVert^2\sum_{j=0}^{n-1}\lVert a_j\rVert^2\right)^{\frac{1}{2}}.

Corollary 2.9. (Noncommutative Carmichael-Mason Lower Bound) Let $\mathcal A$ be a unital Banach algebra. Let

p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in\mathcal A[z],\qquad a_0\text{ and }a_n\text{ are invertible}.

Let $u\in\mathcal A$ be such that $p(u)=0$ , $u$ is invertible and

\lVert(u^{-1})^n\rVert=\lVert u^{-1}\rVert^n.

Then

\frac{1}{\lVert u^{-1}\rVert}>\frac{1}{\left(1+\sum_{j=1}^{n}\lVert a_0^{-1}a_j\rVert^2\right)^{\frac{1}{2}}}\ge\frac{1}{\left(1+\lVert a_0^{-1}\rVert^2\sum_{j=1}^{n}\lVert a_j\rVert^2\right)^{\frac{1}{2}}}.

Conclusions

Over two centuries ago, Lagrange established an upper bound for the zeros of complex polynomials [10, 13]. In 1916, Fujiwara found this bound independently [5].
In 1914, Carmichael and Mason derived an upper bound for the zeros of complex polynomials [3].
In 1916, Kuniyeda improved Carmichael-Mason bound [7].
In this article, we derived noncommutative versions of Lagrange-Fujiwara and Kuniyeda bounds.

References

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