Symbolic computation of the roots of any polynomial with integer coefficients

Abstract

The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive' operations like replacement of sequences and counting of symbols. No calculations using 'advanced' operations like multiplication, division, logarithms etc. are needed. The method can be implemented as a geometric construction of roots of polynomials to arbitrary accuracy using only a straight edge, a compass, and pencils of 2m different colors. In particular, the ancient problem of the "doubling of cube" is soluble asymptotically by the above-mentioned construction. This method, by which a cube can be doubled, albeit, in infinite steps, is probably the closest to the original problem of construction using only a straight edge and compass in a finite number of steps. Moreover, to every polynomial of degree m over the field of rationals, can be associated an m-term recurrence relation for generating integer sequences. A set of m such sequences, which together exhibit interesting properties related to the roots of the polynomial, can be obtained if the m initial terms of each of these m sequences is chosen in a special way using a matrix associated with the polynomial. Only two of these integer sequences need to be computed to obtain the real root having the largest absolute value. Since this method involves only integers, it is faster than the conventional methods using floating-point arithmetic.