Euler Characteristics of Real Varieties

J.W. Bruce

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

In this paper we show how to associate to any real projective algebraic variety Z ⊂ RPn−1 a real polynomial F1:Rn,0 → R, 0 with an algebraically isolated singularity, having the property that χ(Z) = ½(1 − deg (grad F1), where deg (grad F1 is the local real degree of the gradient grad F1:Rn, 0 → Rn, 0. This degree can be computed algebraically by the method of Eisenbud and Levine, and Khimshiashvili [5]. The variety Z need not be smooth. This leads to an expression for the Euler characteristic of any compact algebraic subset of Rn, and the link of a quasihomogeneous mapping f: Rn, 0 → Rn, 0 again in terms of the local degree of a gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitrary algebraic subset of Rn and the link of any polynomial map are given in terms of the degrees of algebraically finite gradient maps. These maps do involve ‘sufficiently small’ constants, but the degrees involved ar (theoretically, at least) algebraically computable.
Original languageEnglish
Pages (from-to)547-552
JournalBulletin of the London Mathematical Society
Volume22
Issue number6
DOIs
Publication statusPublished - 1990

Fingerprint

Euler Characteristic
Isolated Singularity
Gradient
Polynomial Maps
Subset
Algebraic Variety
Polynomial
Arbitrary

Cite this

Bruce, J.W. / Euler Characteristics of Real Varieties. In: Bulletin of the London Mathematical Society. 1990 ; Vol. 22, No. 6. pp. 547-552.
@article{a389f8eb009a4bb9841baa4ef3fc3153,
title = "Euler Characteristics of Real Varieties",
abstract = "In this paper we show how to associate to any real projective algebraic variety Z ⊂ RPn−1 a real polynomial F1:Rn,0 → R, 0 with an algebraically isolated singularity, having the property that χ(Z) = ½(1 − deg (grad F1), where deg (grad F1 is the local real degree of the gradient grad F1:Rn, 0 → Rn, 0. This degree can be computed algebraically by the method of Eisenbud and Levine, and Khimshiashvili [5]. The variety Z need not be smooth. This leads to an expression for the Euler characteristic of any compact algebraic subset of Rn, and the link of a quasihomogeneous mapping f: Rn, 0 → Rn, 0 again in terms of the local degree of a gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitrary algebraic subset of Rn and the link of any polynomial map are given in terms of the degrees of algebraically finite gradient maps. These maps do involve ‘sufficiently small’ constants, but the degrees involved ar (theoretically, at least) algebraically computable.",
author = "J.W. Bruce",
year = "1990",
doi = "10.1112/blms/22.6.547",
language = "English",
volume = "22",
pages = "547--552",
journal = "Bulletin of the London Mathematical Society",
issn = "0024-6093",
publisher = "Oxford University Press",
number = "6",

}

Euler Characteristics of Real Varieties. / Bruce, J.W.

In: Bulletin of the London Mathematical Society, Vol. 22, No. 6, 1990, p. 547-552.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Euler Characteristics of Real Varieties

AU - Bruce, J.W.

PY - 1990

Y1 - 1990

N2 - In this paper we show how to associate to any real projective algebraic variety Z ⊂ RPn−1 a real polynomial F1:Rn,0 → R, 0 with an algebraically isolated singularity, having the property that χ(Z) = ½(1 − deg (grad F1), where deg (grad F1 is the local real degree of the gradient grad F1:Rn, 0 → Rn, 0. This degree can be computed algebraically by the method of Eisenbud and Levine, and Khimshiashvili [5]. The variety Z need not be smooth. This leads to an expression for the Euler characteristic of any compact algebraic subset of Rn, and the link of a quasihomogeneous mapping f: Rn, 0 → Rn, 0 again in terms of the local degree of a gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitrary algebraic subset of Rn and the link of any polynomial map are given in terms of the degrees of algebraically finite gradient maps. These maps do involve ‘sufficiently small’ constants, but the degrees involved ar (theoretically, at least) algebraically computable.

AB - In this paper we show how to associate to any real projective algebraic variety Z ⊂ RPn−1 a real polynomial F1:Rn,0 → R, 0 with an algebraically isolated singularity, having the property that χ(Z) = ½(1 − deg (grad F1), where deg (grad F1 is the local real degree of the gradient grad F1:Rn, 0 → Rn, 0. This degree can be computed algebraically by the method of Eisenbud and Levine, and Khimshiashvili [5]. The variety Z need not be smooth. This leads to an expression for the Euler characteristic of any compact algebraic subset of Rn, and the link of a quasihomogeneous mapping f: Rn, 0 → Rn, 0 again in terms of the local degree of a gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitrary algebraic subset of Rn and the link of any polynomial map are given in terms of the degrees of algebraically finite gradient maps. These maps do involve ‘sufficiently small’ constants, but the degrees involved ar (theoretically, at least) algebraically computable.

U2 - 10.1112/blms/22.6.547

DO - 10.1112/blms/22.6.547

M3 - Article

VL - 22

SP - 547

EP - 552

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 6

ER -