A Topological Proof of the Fundamental Theorem of Algebra

Arnold, B. H.

doi:10.2307/2305130

Cited by 18 publications

(7 citation statements)

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“…Expanding the new picture of R, a determinant, by its final column gives a linear combination over H of the expressions X j f (X), X k g(X) above so as to equal R X (f, g), which is formula (2). Remark: Now we are prepared to show that any polynomial of degree n ≥ 1 over a real-closed field F, breaks into irreducible factors of degree 1 and degree 2.…”

Section: Properties Of the Resultantmentioning

confidence: 99%

“…Much of the treatment in Gauss II does not appear 'elementary', but it is argued that everything there can be reduced to first-order clauses and derivations. This should not be possible when carrying through less 'algebraic' proofs of FTA, such as one that relies on Brouwer's fixed-point theorem for a Euclidean ball [2], or on a maximum principle for functions on a compact planar domain [2] (see also texts in complex variables such as [21]). …”

Section: Introductionmentioning

confidence: 99%

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Computable Implementation of ``Fundamental Theorem of Algebra"

Sjogren¹,

Li²,

Zhao³

et al. 2013

Int. J. of Pure and Appl. Math.

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The Fundamental Theorem of Algebra (FTA) has been studied for more than 300 years: more or less satisfactory proofs of FTA emerged in the 18th and 19th centuries. Proofs denoted as 'algebraic' or 'elementary' derived from the axioms defining a Real-Closed Field (RCF). A proof is given that brings up-to-date work of Gauss (1816) and P. Gordan (1879). It does not refer explicitly to the complex numbers but instead works with auxiliary polynomials in two variables. We report that computer software has been developed to effect symbolic calculation in the context of exact arithmetic. Some examples show how these routines apply to the algebra of symmetric multinomial forms used in Laplace's proof (1795) of FTA, as well as to the theory of Sylvester forms and the Bézoutian formulation of the resultant.

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Section: Properties Of the Resultantmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computable Implementation of ``Fundamental Theorem of Algebra"

Sjogren¹,

Li²,

Zhao³

et al. 2013

Int. J. of Pure and Appl. Math.

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“…The above procedure is only guaranteed to obtain a single root γ 1 . Subsequent roots may be estimated by taking f (1) (z) ≡ f (z)(1 − z/γ 1 ) −1 and finding another root γ 2 , then iterating the procedure with f (j) (z) ≡ f (z)(1 − z/γ j ) −1 , j = 1, 2, . .…”

Section: Parallel Numerical Procedures For Finding All Roots Of a Pol...mentioning

confidence: 99%

“…The fundamental theorem of algebra states that any polynomial function from the complex numbers to the complex numbers with complex coefficients has at least one root. There are several proofs of the fundamental theorem of algebra, which employ a number of different domains of mathematics, including complex analysis (Liouville's theorem, Cauchy's integral theorem or the mean value property) ( [9] [11], topology (Brouwer's fixed point theorem) [1], differential topology [6], calculus ( [4], [9]), and "elementary" methods using meshes or lattices [8], [2] (for easily-accessible and readable web references, see [5], [10], [7], [3]). Many of these proofs are beautiful and elegant.…”

Section: Introductionmentioning

confidence: 99%

A Visualizable, Constructive Proof of the Fundamental Theorem of Algebra, and a Parallel Polynomial Root Estimation Algorithm

Thron,

Barry

2020

Preprint

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This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to convey the gist of the proof. A rigorous version of the proof can be developed using only single-variable calculus and basic properties of complex numbers, but the technical details are somewhat involved. In order to facilitate the reader's intuitive grasp of the proof, we first present the main points of the argument, which can be illustrated by computer experiments. Next we fill in some of the details, using single-variable calculus. Finally, we give a numerical procedure for finding all roots of an nth degree polynomial by solving 2n differential equations in parallel.

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“…, x k and i F which satisfy certain integrability conditions [4]. Then it considered that the transformations of i F produces result if one traces their values along a closed loop.…”

Section: Introductionmentioning

confidence: 99%