Bezout theorem

Bezout's theorem on the division of a polynomial by a linear binomial: The remainder of the division of the polynomial

$$f(x)=a_0x^n+\dotsb+a_n$$

by the binomial $x-a$ is $f(a)$. It is assumed that the coefficients of the polynomials are contained in a certain commutative ring with a unit element, e.g. in the field of real or complex numbers. A consequence of Bezout's theorem is the following: A number $\alpha$ is a root of the polynomial $f(x)$ if and only if $f(x)$ is divisible by the binomial $x-\alpha$ without remainder.

Bezout's theorem on homogeneous equations: If a system of $n$ homogeneous equations in $n+1$ unknowns

$$f_i(x_0,\dots,x_n)=0,\quad i=1,\dots,n,\tag{*}$$

has only a finite number of non-proportional non-zero solutions in an algebraically closed field containing the coefficients of the system, then the number of these solutions counted according to their multiplicity is equal to the product of the degrees of the equations. The multiplicity of the solutions is, by definition, the intersection index (in algebraic geometry) of the hypersurfaces \ref{*} at the respective point. The theorem is called after E. Bezout [1], who studied systems of algebraic equations of higher degrees.

References

[1]	E. Bezout, "Théorie génerale des équations algébriques" , Paris (1779)

References

[a1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) pp. Chapt. 4, Sect. 2 (Translated from Russian) MR0447223 Zbl 0362.14001