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Solvability by radicals is in polynomial time

1985
*
Journal of computer and system sciences (Print)
*

A

doi:10.1016/0022-0000(85)90013-3
fatcat:acjvol7675brdgh6tsluvtkjeq
*polynomial**time*algorithm*is*presented for the founding question of Galois theory: determining*solvability**by**radicals*of a manic irreducible*polynomial*over the integers. ... Also a*polynomial**time*algorithm which expresses a root*in**radicals**in*terms of a straightline program*is*given. ... Furthermore, if the*polynomial**is**solvable**by**radicals*, we can express the roots*in**radicals*using a suitable encoding. ...##
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Denesting by bounded degree radicals
[chapter]

1997
*
Lecture Notes in Computer Science
*

The running

doi:10.1007/3-540-63397-9_5
fatcat:eboscpm7oze7xemzmltyzznswe
*times*of the algorithms are*polynomial**in*the description size of the splitting eld for the original nested*radical*. ... Given a nested*radical*involving only d-th roots we show how to compute an optimal or near optimal depth denesting of this nested*radical**by*a nested*radical*that involves only D-th roots, where D*is*an ...*In*particular, it can be decided*in**polynomial**time*whether the extension F i : F i?1*is**solvable**by*order d*radicals*. ...##
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Solving polynomials by radicals with roots of unity in minimum depth

1999
*
Mathematics of Computation
*

Let α be a root of a

doi:10.1090/s0025-5718-99-01060-1
fatcat:ffimh4x33necji62u24pcgvpfm
*polynomial*f ∈ k[x] which*is**solvable**by**radicals*. Let L be the splitting field of α over k. ... We show that an optimal nested*radical*with roots of unity for α can be effectively constructed from the derived series of the*solvable*Galois group of L(ζn) over k(ζn). ... Introduction It was shown*in*[8] that whether a*polynomial*with rational coefficients*is**solvable**by**radicals*can be decided*in**polynomial**time*. ...##
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Limitations to algorithm solvability: Galois methods and models of computation

1986
*
Proceedings of the fifth ACM symposium on Symbolic and algebraic computation - SYMSAC '86
*

x», where Q

doi:10.1145/32439.32453
dblp:conf/issac/Bajaj86
fatcat:okyzs77xyndizicj3276dcbm4a
*is*the field of rationals and q(x)eQ [x] are*polynomials*with non-*solvable*Galois groups. ...*In*particular we show that lhere exist applied computational problems for which there are no closed form solutions over models such as Q(+,", .,/, v), Q (+, _, '10, /, tv), and Q(+, -, '1<,/, k...J, q( ... Introduction The now well known Theorem of Abel-Ruffini, (proved*by*Ruffini*in*1813 and independently*by*Abel*in*1827) states that the •general' equation of n lh degree, (n~),*is*not*solvable**by**radicals*...##
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The algebraic degree of geometric optimization problems

1988
*
Discrete & Computational Geometry
*

*In*particular we show that the classic Weber problem, along with the line-restricted Weber problem and its three-dimensional version are

*in*general not

*solvable*

*by*

*radicals*over the field of rationals. ...

*In*this paper we apply Galois methods to certain fundamental geometric optimization problems whose exact computational complexity has been an open problem for a long

*time*. ... Acknowledgments Sincere thanks to John Hopcroft for his suggestions

*in*the use of algebraic methods and to Walter Schnyder for his explanations on the intricacies of logic. ...

##
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Commentary on Ballou's paper: Galois - The Myths and the Man

2005
*
The Mathematics Enthusiast
*

Since S 5

doi:10.54870/1551-3440.1020
fatcat:gagf5urc7zbqtekdoojfhkfzve
*is*not*solvable*, then f (x)*is*not*solvable*. Therefore, there exists a*polynomial*of degree 5 that*is*not*solvable*algebraically*by**radicals*6 . ... Galois had answered the essential question of what m akes a*polynomial**solvable**by**radicals*. Understanding Galois theory*is*not an easy task. ...##
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A Complete Invariant Generation Approach for P-solvable Loops
[chapter]

2010
*
Lecture Notes in Computer Science
*

Our experimental results report that our method takes less iterations and/or

doi:10.1007/978-3-642-11486-1_21
fatcat:swfitcpnhnervkr6e6apuws56q
*time*than other*polynomial*invariant generation techniques. ... We present an algorithm for generating all*polynomial*invariants of Psolvable loops with assignments and nested conditionals. We prove termination of our algorithm. ... (i) First, the body of a P-*solvable*loop*is*described*by*recurrence equations*in*the loop counter. ...##
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Seismic Solvability Problems
[article]

2012
*
arXiv
*
pre-print

We can classify

arXiv:1212.1350v1
fatcat:ithynhp6obaxhop4a2orlo77zm
*polynomials*into simple (*solvable**by**radicals*) and complex (not*solvable**by**radicals*). ... If we define*solvability**by*using only square roots, cube roots etc, then*polynomials*are not*solvable**by**radicals*(square root, cube root etc). ... (Complex numbers) Fundamental theorem of algebra (Gauss)*Solvable**by**radicals*(Abel) Group theory ( ...##
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A Note on the Unsolvability of the Weighted Region Shortest Path Problem
[article]

2013
*
arXiv
*
pre-print

*In*the ACMQ, one can compute exactly any number that can be obtained from the rationals Q

*by*applying a finite number of operations from +, -, ×, , √(), for any integer k >= 2. ... The weighted region shortest path problem

*is*to determine a shortest path

*in*S between two points s, t

*in*R^2, where the distances are measured according to the weighted Euclidean metric-the length of ... When the degree of the

*polynomial*equations involved

*in*the solution of a problem

*is*unbounded, then an unsolvability result like the one presented

*in*this paper justifies the search for an approximate ...

##
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How ( Δt )5 + A1 ⋅ ( Δt )2 + A2 = 0 Is Generally, in the Galois Sense Solvable for a Kerr-Newman Black Hole Affect Questions on the Opening and Closing of a Wormhole Throat and the Simplification of the Problem, Dramatically Speaking, If d = 1 (Kaluza Klein Theory) and Explaining the Lack of Overlap with the Results When Applying the Gauss-Lucas Theorem

2019
*
Journal of High Energy Physics Gravitation and Cosmology
*

how the physics answers are all

doi:10.4236/jhepgc.2019.51014
fatcat:ffoq7ktwxnh7xbn74di74wqusu
*radically*different for abbreviated lower math tech answers to this problem. i.e. if one turns the 1 2 ⋅*Is*Generally,*in*the Galois Sense*Solvable*for a Kerr-Newman Black ... First off, the term t ∆*is*for the smallest unit of*time*step. ... Here,*polynomial*f(x) = 0*is**solvable**by**radicals*, means that definitions as to*solvability**in*[80]*is*satisfied*in*that we have operations given*in*the examples delineated*by*[81] . ...##
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THE COMPLEXITY OF THE EQUIVALENCE PROBLEM OVER FINITE RINGS

2011
*
Glasgow Mathematical Journal
*

We prove that for a finite ring if the factor

doi:10.1017/s001708951100053x
fatcat:wsaerkkpjndjffq35h543ylfla
*by*the Jacobson*radical*can be lifted*in*the centre, then this problem can be solved*in**polynomial**time*. ... They proved that for finite nilpotent rings the*polynomial*equivalence problem could be solved*in**polynomial**time**in*the length of the two input*polynomials*. ... Let R be a finite ring and J be its Jacobson*radical*. If R/J*is*commutative, then the sigma equivalence problem for R*is**solvable**in**polynomial**time*. ...##
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Polynomial time algorithms for sentences over number fields

1992
*
Information and Computation
*

We also show that ther are

doi:10.1016/0890-5401(92)90037-g
fatcat:zzycxn2c6rbf3e6mky7ttczhqy
*polynomial**time*algorithms to decide whether or not cp*is*true*in*every algebraic number field or every*radical*extension field of Q. 0 1992 Academic Press, Inc. ... We show that given an arbitrary algebraic number field K there*is*a*polynomial**time*algorithm to decide whether rp*is*true*in*K or not. ...*Solvability**by**radicals**is**in**polynomial**time*(Landau and Miller, 1985) . ...##
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Practical approach to solvability: Geophysical application using complex decomposition into simple part (solvable) and complex part (interpretable) for seismic imaging
[article]

2010
*
arXiv
*
pre-print

The simple part

arXiv:1012.0520v1
fatcat:e37keih2h5aezcps6zoi5rwnci
*is*the*solvable*part*by*the method prescribed*in*the problem definition. The complex part*is*the leftover of the simple part. ... The classical approach to*solvability*of a mathematical problem*is*to define a method which includes certain rules of operation or algorithms. ... The quadratic*is**solvable**by**radicals*. But the higher order*polynomials*are not*solvable**by**radicals**in*general. ...##
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Upper Bounds on the Complexity of Some Galois Theory Problems
[chapter]

2003
*
Lecture Notes in Computer Science
*

For

doi:10.1007/978-3-540-24587-2_73
fatcat:kt3yapvwqfej7ay7mt5kk6dig4
*polynomials*f with*solvable*Galois group we show that the order can be computed exactly*by*a randomized*polynomial*-*time*algorithm with access to an NP oracle. ... Furthermore, the order can be approximated*by*a randomized*polynomial*-*time*algorithm with access to an NP oracle. ...*In*fact, we show that for*solvable*Galois groups, finding |G|*is**polynomial*-*time*reducible to approximating |G|. To begin with we need a test for*solvability**by**radicals*. ...##
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Radically solvable graphs
[article]

2012
*
arXiv
*
pre-print

It

arXiv:1207.1580v1
fatcat:djwobc5dfrhj7argnqiody2mbm
*is**radically**solvable*if the set of vertex coordinates*is*contained*in*a*radical*extension of the field of rationals extended*by*the squared edge lengths. ... A 2-dimensional framework*is*a straight line realisation of a graph*in*the Euclidean plane. ... Then I 2n+1*is*generated*by*a single*polynomial*h 2n+1 ∈ K[X 2n+1 ], and every zero of h 2n+1*in*C extends to a zero of I*in*C 2n+1*by*Lemma 3.7. ...
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