Bézout's theorem for curves states that, in general, two algebraic curves of degrees
and intersect in points and cannot meet in more than points unless they have a component in common (i.e.,
the equations defining them have a common factor; Coolidge 1959, p. 10).
Bézout's theorem for polynomials states that if and are two polynomials with no
roots in common, then there exist two other polynomials and such that . Similarly, given polynomial equations of degrees
, , ... in variables, there are in general common solutions.
Séroul (2000, p. 10) uses the term Bézout's theorem for the following two theorems.
1. Let
be any two integers, then there exist such that
2. Two integers
and
are relatively prime if there exist such that
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 10, 1959.Séroul,
R. "The Bézout Theorem." §2.4.1 in Programming
for Mathematicians. Berlin: Springer-Verlag, p. 10, 2000.Shub,
M. and Smale, S. "Complexity of Bézout's Theorem. I. Geometric Aspects."
J. Amer. Math. Soc.6, 459-501, 1993.Shub, M. and Smale,
S. "Complexity of Bézout's Theorem. II. Volumes and Probabilities."
In Computational Algebraic Geometry (Nice, 1992). Boston, MA: Birkhäuser,
pp. 267-285, 1993.Shub, M. and Smale, S. "Complexity of Bézout's
Theorem. III. Condition Number and Packing." J. Complexity9,
4-14, 1993.Shub, M. and Smale, S. "Complexity of Bézout's
Theorem. IV. Probability of Success; Extensions." SIAM J. Numer. Anal.33,
128-148, 1996.Shub, M. and Smale, S. "Complexity of Bézout's
Theorem. V. Polynomial Time." Theoret. Comput. Sci.134, 141-164,
1994.
and 
intersect in 
points and cannot meet in more than 
points unless they have a component in common (i.e.,
the equations defining them have a common factor; Coolidge 1959, p. 10).
and 
are two polynomials with no
roots in common, then there exist two other polynomials 
and 
such that 
. Similarly, given 
polynomial equations of degrees
, 
, ...
in 
variables, there are in general 
common solutions.
be any two integers, then there exist 
such that
and 
are relatively prime if there exist 
such that
