Bounds on absolute positiveness of multivariate polynomials

Quality of positive root bounds

“…3 In fact, we prove a more general result: we show that every positive root bound which is also an absolute positiveness bound (a bound on the largest positive root and the positive roots of the derivatives, see Definition 1) can be arbitrarily bad. All positive root bounds listed at the beginning of this paper are absolute positiveness bounds, as well as every positive root bound derived in the framework in [2]. It also appears that every positive root bound derived in the framework in [1] is an absolute positiveness bound, although we do not have a proof.…”

“…As a consequence, there has been intensive effort on finding such bounds. (see [7,12,1,5,2,4,8,10,6] for some examples).…”

“…An example in Theorem 5.3 of [5] shows that the relative over-estimations of B L , B C , and B K can approach infinity, even when the degree is fixed and when the number of sign variations is the same as the number of positive roots. The proof strategy for Theorem 2 can be easily adapted to show that the relative over-estimation of every positive root bound in the framework in [2] is at most linear in the degree. It also immediate that there exists at least one positive root bound (namely B K ) in the framework from [1] whose relative over-estimation can approach infinity.…”

“…It is not obvious that B H can be computed in linear time since it involves max over min. However, Melhorn and Ray[11] found an ingenious way to compute it in linear time 2. Stefanescu, Akritas, Strzebonksi, Vigklas, Batra and Sharma[12,1,2] extended the above bounds by splitting single monomials as sums of several monomials and considering different groupings of positive and negative monomials.…”

“…However, Melhorn and Ray[11] found an ingenious way to compute it in linear time 2. Stefanescu, Akritas, Strzebonksi, Vigklas, Batra and Sharma[12,1,2] extended the above bounds by splitting single monomials as sums of several monomials and considering different groupings of positive and negative monomials. Batra and Sharma[2] showed that the tightest bound in their framework improves on B H by at most a constant.…”