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Trinomials and Deterministic Complexity Limits for Real Solving

Emma Boniface, Weixun Deng, J. Maurice Rojas

TL;DR

This work advances deterministic real-root solving for sparse univariate polynomials by focusing on trinomials f(x)=c_1+c_2 x^{a_2}+c_3 x^{a_3}. It introduces an ill-conditioning criterion tied to a trinomial discriminant and proves that, for the non-ill-conditioned majority of inputs, one can compute all real roots to Smale-approximate accuracy in time \\log^{4+o(1)}(dH), with each approximate root having height O(\\log(dH)). The methodology blends Baker-type bounds on linear forms in logarithms, \mathcal{A}-hypergeometric series to seed Newton iterations, and a careful Newton-analytic framework via a global parameter Γ_f to certify convergence. The paper also links real-root solving to Koiran's Trinomial Sign Problem, showing a substantial fraction of inputs admit polynomial-time sign evaluation at rational points when the trinomial is not ill-conditioned. Finally, it discusses a root-separation chasm at four terms, illustrating why efficient deterministic real-solving is plausible for binomials and trinomials but may break down for tetranomials, and it sketches potential extensions to circuit systems and higher-dimensional sparse problems.

Abstract

We detail an algorithm that -- for all but a $\frac{1}{Ω(\log(dH))}$ fraction of $f\in\mathbb{Z}[x]$ with exactly $3$ monomial terms, degree $d$, and all coefficients in $\{-H,\ldots, H\}$ -- 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 $f\in\mathbb{Z}[x]$ with coefficients in $\{-H,\ldots,H\}$, at a point $r\!\in\!\mathbb{Q}$ 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 $1-\frac{1}{Ω(\log(dH))}$ of the inputs $(f,r)$.

Trinomials and Deterministic Complexity Limits for Real Solving

TL;DR

This work advances deterministic real-root solving for sparse univariate polynomials by focusing on trinomials f(x)=c_1+c_2 x^{a_2}+c_3 x^{a_3}. It introduces an ill-conditioning criterion tied to a trinomial discriminant and proves that, for the non-ill-conditioned majority of inputs, one can compute all real roots to Smale-approximate accuracy in time \\log^{4+o(1)}(dH), with each approximate root having height O(\\log(dH)). The methodology blends Baker-type bounds on linear forms in logarithms, \mathcal{A}-hypergeometric series to seed Newton iterations, and a careful Newton-analytic framework via a global parameter Γ_f to certify convergence. The paper also links real-root solving to Koiran's Trinomial Sign Problem, showing a substantial fraction of inputs admit polynomial-time sign evaluation at rational points when the trinomial is not ill-conditioned. Finally, it discusses a root-separation chasm at four terms, illustrating why efficient deterministic real-solving is plausible for binomials and trinomials but may break down for tetranomials, and it sketches potential extensions to circuit systems and higher-dimensional sparse problems.

Abstract

We detail an algorithm that -- for all but a fraction of with exactly monomial terms, degree , and all coefficients in -- produces an approximate root (in the sense of Smale) for each real root of in deterministic time in the classical Turing model. (Each approximate root is a rational with logarithmic height .) The best previous deterministic bit complexity bounds were exponential in . We then relate this to Koiran's Trinomial Sign Problem (2017): Decide the sign of a degree trinomial with coefficients in , at a point of logarithmic height , in (deterministic) time . We show that Koiran's Trinomial Sign Problem admits a positive solution, at least for a fraction of the inputs .
Paper Structure (9 sections, 12 theorems, 12 equations)

This paper contains 9 sections, 12 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. The Root Separation Chasm at Four Terms
  3. Going Beyond Univariate Trinomials
  4. Background
  5. Approximating Logarithms and Roots of Binomials
  6. Discriminants and $\alpha$-Theory for Trinomials
  7. $\mathcal{A}$-Hypergeometric Functions
  8. Solving Trinomial Equations over $\mathbb{R}$
  9. Signs and Solving are Roughly Equivalent for Trinomials

Key Result

Theorem 1.1

Following the notation above, assume $f$ is not ill-conditioned. Then we can find, in deterministic time $\log^{4+o(1)}(dH)$, a set $\left\{\frac{r_1}{s_1},\ldots,\frac{r_m}{s_m}\right \}\!\subset\!\mathbb{Q}$ of cardinality $m\!=\!m(f)$ such that: 1. For all $j$ we have $r_j\!\neq\!0 \Longrightarro

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 1.8
  • Conjecture 1.9
  • Theorem 2.1
  • ...and 10 more