DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS

Finally, we prove the Definable connectedness conjecture and hence the Pierce-Birkhoff conjecture for an arbitrary regular 2-dimensional local ring A; we also prove Conjecture

It turns out that we can compute these invariants within the same complexity (avoiding further- more any univariate irreducibility test) for a larger class of polynomial: we say that

For the study of multiple roots of sparse polynomials and, in particular, to prove Theorem 2.6, it is not enough to dispose of a version of Theorem 2.2 with an explicit dependence

The fact that a real univariate polynomial misses some real roots is usually over- came by considering complex roots, but the price to pay for, is a complete lost of the sign

Finally, we prove the Definable connectedness conjecture and hence the Pierce-Birkhoff conjecture for an arbitrary regular 2-dimensional local ring A; we also prove Conjecture

In Section 3, we detail a fast algorithm for the computation of affine factors , which are polynomials having the same roots as the shifts but which can be computed more

• In Algorithm 2, if the degree d of the polynomial Q satisfies d ≤ 4, its roots are directly computed using arithmetic expressions and the rounding errors affecting them are

A’Campo and Oka describe the embedded resolution of a plane branch by a sequence of toric modifications using approximate roots in [A’C-Ok] and give topological proofs of some of