Mason–Stothers theorem
Encyclopedia
The Mason–Stothers theorem, or simply Mason's theorem, is a mathematical theorem
about polynomials, analogous to the abc conjecture
for integers. It is named after W. Wilson Stothers, who published it in 1981, and R. C. Mason, who rediscovered it shortly thereafter.
The theorem states:
Let , , and be relatively prime polynomials such that , with coefficients that are either real number
s or complex number
s. Then
where is the polynomial of minimum degree that has the same roots as , so gives the number of distinct roots of .
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
about polynomials, analogous to the abc conjecture
Abc conjecture
The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c , which have no common factor and satisfy a + b = c...
for integers. It is named after W. Wilson Stothers, who published it in 1981, and R. C. Mason, who rediscovered it shortly thereafter.
The theorem states:
Let , , and be relatively prime polynomials such that , with coefficients that are either real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s or complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. Then
where is the polynomial of minimum degree that has the same roots as , so gives the number of distinct roots of .
External links
- Mason-Stothers Theorem and the ABC Conjecture, Vishal Lama. A cleaned-up version of the proof from Lang's book.