Remarks on the Gauss-Lucas theorem in higher dimensional space

Author:
A. W. Goodman

Journal:
Proc. Amer. Math. Soc. **55** (1976), 97-102

MSC:
Primary 30A08; Secondary 26A78

DOI:
https://doi.org/10.1090/S0002-9939-1976-0435366-6

MathSciNet review:
0435366

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Abstract | References | Similar Articles | Additional Information

Abstract: A recent paper by J. B. Diaz and Dorothy Browne Shaffer extends the Gauss-Lucas Theorem to -dimensional Euclidean space. The authors leave open certain natural questions concerning the existence of ``zeros of the derivative". This paper answers three such questions, and suggests several other questions for further investigation.

**[1]**J. B. Diaz and Dorothy Browne Shaffer,*A generalization to higher dimensions of a theorem of Lucas concerning the zeros of the derivative of a polynomial of one complex variable*, Proc. Internat. Congress Math. (Vancouver, 1974), Canad. Math. Congress (to appear). MR**0435365 (55:8325)****[2]**N. Dunford and J. T. Schwartz,*Linear operators*. I:*General theory*, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR**22**#8302. MR**0117523 (22:8302)****[3]**Morris Marden,*Geometry of polynomials*, Math. Surveys, no. 3, Amer. Math. Soc., Providence, R.I., 1966. MR**27**#1562. MR**0225972 (37:1562)****[4]**G.V. Sz.-Nagy,*Über die Lage der Nullstellen eines Abstandspolynoms and seiner Derivierten*, Bull. Amer. Math. Soc.**55**(1949), 329-342. MR**10**, 702. MR**0030036 (10:702g)****[5]**A. Schurrer,*On the location of the zeros of the derivative of rational functions of distance polynomials*, Trans. Amer. Math. Soc.**89**(1958), 100-112. MR**20**#4634. MR**0098172 (20:4634)**

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0435366-6

Keywords:
Polynomials,
zeros of the derivative,
Euclidean -space,
gradient,
Gauss-Lucas Theorem

Article copyright:
© Copyright 1976
American Mathematical Society