Michael Sagraloff
ABSTRACT
Let p ∈ Z[x] be an arbitrary polynomial of degree n with k non-zero integer coefficients of absolute value less than 2τ. 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(k3·log(nτ)·logn) many exact arithmetic operations over the rational numbers.
When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by Õ(k4·n·(τ+k)), where Õ(·) indicates the omission of logarithmic factors. Hence, for sufficiently sparse polynomials (i.e. k = O(logc(nτ)) for a constant c), the bit complexity is Õ(nτ), which is optimal up to logarithmic factors.
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Index Terms
- A near-optimal algorithm for computing real roots of sparse polynomials
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