| Abstrakt: |
In 1917 Pell1 and Gordon used sylvester2, Sylvester's little known and hardly ever used matrix of 1853, to compute2 the coefficients of a Sturmian remainder -- obtained in applying in ℚ[x], Sturm's algorithm on two polynomials f, g ∈ ℤ[x] of degree n -- in terms of the determinants3 of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900.4 In this paper we extend the work by Pell and Gordon and show how to compute2 the coefficients of an Euclidean remainder -- obtained in finding in ℚ[x], the greatest common divisor of f, g ∈ ℤ[x] of degree n -- in terms of the determinants5 of the corresponding submatrices of sylvester1, Sylvester's widely known and used matrix of 1840. [ABSTRACT FROM AUTHOR] |
