Root separation for trinomials

Koiran, Pascal

doi:10.1016/j.jsc.2019.02.004

Cited by 11 publications

(9 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Section: Introductionsupporting

confidence: 67%

“…Our complexity bound from Theorem 1.1 appears to be new, and complements earlier work on the arithmetic complexity of approximating [27,29] and counting [5,14] real roots of trinomials. In particular, Theorem 1.1 nearly settles a question of Koiran from [14] on the bit complexity of solving trinomial equations over the reals. One should also observe that the best general bit complexity bounds for solving real univariate polynomials are super-linear in d and work in terms of ε-approximation, thus requiring an extra parameter depending on root separation (which is not known a priori): see, e.g., [18,22].…”

Section: Introductionsupporting

confidence: 67%

See 1 more Smart Citation

Trinomials and Deterministic Complexity Limits for Real Solving

Boniface¹,

Deng²,

Rojas³

2022

Preprint

View full text Add to dashboard Cite

We detail an algorithm that -for all but a 1 Ω(log(dH)) fraction of f ∈ Z[x] with exactly 3 monomial terms, degree d, and all coefficients in {−H, . . . , H} -produces an approximate root (in the sense of Smale) for each real root of f in deterministic time log 4+o(1) (dH) in the classical Turing model. (Each approximate root is a rational with logarithmic height O(log(dH)).) The best previous deterministic bit complexity bounds were exponential in log d. We then relate this to Koiran's Trinomial Sign Problem (2017): Decide the sign of a degree d trinomial f ∈ Z[x] with coefficients in {−H, . . . , H}, at a point r ∈ Q of logarithmic height log H, in (deterministic) time log O(1) (dH). We show that Koiran's Trinomial Sign Problem admits a positive solution, at least for a fraction 1 − 1 Ω(log(dH)) of the inputs (f, r).

show abstract

Section: Introductionsupporting

confidence: 67%

Section: Introductionsupporting

confidence: 67%

Trinomials and Deterministic Complexity Limits for Real Solving

Boniface¹,

Deng²,

Rojas³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…On a related matter, Koiran recently used analytic arguments (Rolle's theorem and Baker's theory of linear forms in the logarithms of algebraic numbers) to give a bound on the (classical) root separation for trinomials, with a very small dependency on the degree [7]. It is not clear to us whether similar results also hold for the absolute separation.…”

Section: Discussionmentioning

confidence: 99%

Absolute Root Separation

Bugeaud

Dujella

Fang³

et al. 2019

Experimental Mathematics

View full text Add to dashboard Cite

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic. Separation and absolute separationThe absolute separation of a polynomial P ∈ C[X] is the minimal nonzero distance between the absolute values of its complex roots:Having good lower bounds on this quantity for polynomials with integer coefficients is of interest in the asymptotic analysis of linear recurrent sequences. Before this work, the best bounds that we are aware of [6,3] are of the form abs sep(P ) ≫ H(P ) −d(d 2 +2d−1)/2 , where, here and below, the constant implicit in the ≫ sign depends only on the degree d, while H(P ), the height of the polynomial P , is the maximum of the absolute values of its coefficients. In this work, we improve the exponent of this bound and that of related problems. Still, we do not know how far the exponent we obtain is from being optimal. Thus an important part of this article is devoted to experiments in low degree, from where we can infer families of polynomials exhibiting a behaviour in H(P ) −d−1 for d ∈ {3, 4, 5, 6}.In the much more classical case of the separation sep(P ) := min P (α)=P (β)=0, α =β |α − β|,

show abstract

“…We prove Theorem 1.6 in Section 3. Theorem 1.6 provides a p-adic analogue of a separation bound of Koiran for complex roots of trinomials [34]. As to whether our lower bound is optimal, there are recent examples from [25] showing that log |ζ 1 − ζ 2 | p = −Ω(log max{d, H}) can occur.…”

Section: Introductionmentioning

confidence: 95%

Root Repulsion and Faster Solving for Very Sparse Polynomials Over $p$-adic Fields

Rojas¹,

Zhu²

2021

Preprint

View full text Add to dashboard Cite

For any fixed field K ∈ {Q 2 , Q 3 , Q 5 , . . .}, we prove that all polynomials f ∈ Z[x] with exactly 3 (resp. 2) monomial terms, degree d, and all coefficients having absolute value at most H, can be solved over K within deterministic time log 4+o(1) (dH) log 3 (d) (resp. log 2+o(1) (dH)) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of f in K, and for each such root generates an approximation in Q with logarithmic height O(log 2 (dH) log(d)) that converges at a rate of O (1/p) 2 i after i steps of Newton iteration. We also prove significant speed-ups in certain settings, a minimal spacing bound of p −O(p log 2 p (dH) log d) for distinct roots in C p , and even stronger repulsion when there are nonzero degenerate roots in C p : p-adic distance p −O(log p (dH)) . On the other hand, we prove that there is an explicit family of tetranomials with distinct nonzero roots in Z p indistinguishable in their first Ω(d log p H) most significant base-p digits. Contentsp 2.4. Bit Complexity Basics and Counting Roots of Binomials 2.5. Trees and Roots in Z/(p k ) and Z p 2.6. Trees and Extracting Digits of Radicals 3.

show abstract

Root separation for trinomials

Cited by 11 publications

References 15 publications

Trinomials and Deterministic Complexity Limits for Real Solving

Trinomials and Deterministic Complexity Limits for Real Solving

Absolute Root Separation

Root Repulsion and Faster Solving for Very Sparse Polynomials Over $p$-adic Fields

Contact Info

Product

Resources

About