On the affine Bezout inequality. (English) Zbl 0861.14001

The aim of the paper is to give an elementary proof of the so-called affine Bézout inequality: “The set in \(\overline k^n\) of all common zeroes of the polynomials \(f_1, \dots, f_n \in k[X_1, \dots, X_n]\) is either infinite or has at most \((\deg f_1) \dots (\deg f_n)\) elements \((\overline k\) indicating the algebraic closure of the field \(k)\)”.


14A05 Relevant commutative algebra
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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[1] [CGH] CANIGLIA, L., GALLIGO, A., HEINTZ, J.:Some new effectivity bounds in computational geometry, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proc. 6th Intern. Conf. AAECC-6, Lecture Notes in Computer Science 357, Springer 1989, 131–151
[2] [Ha] HARTSHORNE, R.:Algebraic Geometry Springer 1977
[3] [He] HEINTZ, J.:Definability and fast quantifier elimination over algebraically closed fields, Theoret. Comput. Sci. 24 (1983),239–277 · Zbl 0546.03017
[4] [K] KUNZ, E.:Einführung in die kommutative Algebra und algebraische Geometrie, Vieweg 1980 · Zbl 0432.13001
[5] [R] RENEGAR, J.:On the computational complexity and geometry of the first-order theory of the reals, Part I, School of Operations Research and Industrial Engineering, Cornell University, Technical Report No. 853 (1989)
[6] [RW] RUSEK, K., WINIARSKI, T.: Polynomial Automorphisms of \(\mathbb{C}\) n , Universitis Iacellonicae Acta Mathematica (1984), 143–149 · Zbl 0551.32020
[7] [S] SHAFAREVICH, I.R.:Basic Algebraic Geometry, Grundlehren der math. Wissenschaften 213, Springer 1974
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