Upper Bounds on the Complexity of Some Galois Theory Problems

Arvind, Vikraman; Kurur, Piyush P.

doi:10.1007/978-3-540-24587-2_73

Cited by 4 publications

(10 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also ensure that 𝑝 is such that we can deterministically find the representations 𝛼 1 , . 𝑎 𝑖 (5) and 𝑏 is a solution of (4) .…”

Section: Stepmentioning

confidence: 99%

“…. , 𝑎 𝑘 ) defined in (5). Then (i) 𝑝 is prime with probability at least 1 6𝑠 3 assuming GRH, and (ii) given that 𝑝 is prime, the probability that it divides 𝑁 (𝛼) is at most 2 −𝑠 3 unconditionally.…”

Section: Stepmentioning

confidence: 99%

“…Central to such identity questions on algebraic numbers is the knowledge of the Galois group of the extension where the numbers live in. Algorithmic complexity of computing with Galois groups is investigated in [5,39,46].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Identity Testing for Radical Expressions

Balaji

Nosan

Shirmohammadi

et al. 2022

Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science

View full text Add to dashboard Cite

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit over integers representing a multivariate polynomial 𝑓 (𝑥 1 , . . . , 𝑥 𝑘 ) and nonnegative integers 𝑎 1 , . . . , 𝑎 𝑘 and 𝑑 1 , . . . , 𝑑 𝑘 , written in binary, test whether the polynomial vanishes at the real radicalsWe place the problem in coNP assuming the Generalised Riemann Hypothesis (GRH), improving on the straightforward PSPACE upper bound obtained by reduction to the existential theory of reals. Next we consider a restricted version, called 2-RIT, where the radicals are square roots of prime numbers, written in binary. It was known since the work of Chen and Kao [16] that 2-RIT is at least as hard as the polynomial identity testing problem, however no better upper bound than PSPACE was known prior to our work. We show that 2-RIT is in coRP assuming GRH and in coNP unconditionally. Our proof relies on theorems from algebraic and analytic number theory, such as the Chebotarev density theorem and quadratic reciprocity. CCS CONCEPTS• Mathematics of computing → Probabilistic algorithms; • Computing methodologies → Algebraic algorithms; Number theory algorithms.

show abstract

“…We also ensure that 𝑝 is such that we can deterministically find the representations 𝛼 1 , . 𝑎 𝑖 (5) and 𝑏 is a solution of (4) .…”

Section: Stepmentioning

confidence: 99%

Section: Stepmentioning

confidence: 99%

See 1 more Smart Citation

Identity Testing for Radical Expressions

Balaji

Nosan

Shirmohammadi

et al. 2022

Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science

View full text Add to dashboard Cite

show abstract

“…Indeed enumerating all subfields, regardless of the algorithm used, could not lead to a polynomial time algorithm since the number of subfields is not polynomially bounded, as the example of multi-quadratic fields shows. One may consider a pure Galois-theoretic approach, but it is currently not known whether one can compute in polynomial time, given a number field F , the Galois group of the Galois closure of F (see [1,17,12]). Our method relies on the following Lemma.…”

Section: The Maximal CM Subfieldmentioning

confidence: 99%

Computing groups of Hecke characters

Molin¹,

Page²

2022

Preprint

View full text Add to dashboard Cite

We describe algorithms to represent and compute groups of Hecke characters. We make use of an idèlic point of view and obtain the whole family of such characters, including transcendental ones. We also show how to isolate the algebraic characters, which are of particular interest in number theory. This work has been implemented in Pari/GP, and we illustrate our work with a variety of explicit examples using our implementation.

show abstract

Section: Introductionmentioning

confidence: 99%

Identity Testing for Radical Expressions

Balaji¹,

Nosan²,

Shirmohammadi³

et al. 2022

Preprint

View full text Add to dashboard Cite

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a multivariate polynomial f (x 1 , . . . , x k ) and nonnegative integers a 1 , . . . , a k and d 1 , . . . , d k , written in binary, test whether the polynomial vanishes at the real radicalsWe place the problem in coNP assuming the Generalised Riemann Hypothesis (GRH), improving on the straightforward PSPACE upper bound obtained by reduction to the existential theory of reals. Next we consider a restricted version, called 2-RIT, where the radicals are square roots of prime numbers, written in binary. It was known since the work of Chen and Kao [16] that 2-RIT is at least as hard as the polynomial identity testing problem, however no better upper bound than PSPACE was known prior to our work. We show that 2-RIT is in coRP assuming GRH and in coNP unconditionally. Our proof relies on theorems from algebraic and analytic number theory, such as the Chebotarev density theorem and quadratic reciprocity.

show abstract

Upper Bounds on the Complexity of Some Galois Theory Problems

Cited by 4 publications

References 12 publications

Identity Testing for Radical Expressions

Identity Testing for Radical Expressions

Computing groups of Hecke characters

Identity Testing for Radical Expressions

Contact Info

Product

Resources

About