Let pβZ[x] be an arbitrary polynomial of degree n with k
non-zero integer coefficients of absolute value less than 2Ο. In this
paper, we answer the open question whether the real roots of p can be
computed with a number of arithmetic operations over the rational numbers that
is polynomial in the input size of the sparse representation of p. More
precisely, we give a deterministic, complete, and certified algorithm that
determines isolating intervals for all real roots of p with
O(k3β log(nΟ)β logn) many exact arithmetic operations over the
rational numbers.
When using approximate but certified arithmetic, the bit complexity of our
algorithm is bounded by O~(k4β nΟ), where O~(β )
means that we ignore logarithmic. Hence, for sufficiently sparse polynomials
(i.e. k=O(logc(nΟ)) for a positive constant c), the bit complexity is
O~(nΟ). We also prove that the latter bound is optimal up to
logarithmic factors