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Solvability by radicals is in polynomial time

Susan Landau, Gary Lee Miller
1985 Journal of computer and system sciences (Print)  
A 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.  ... 
doi:10.1016/0022-0000(85)90013-3 fatcat:acjvol7675brdgh6tsluvtkjeq

Denesting by bounded degree radicals [chapter]

Johannes Blömer
1997 Lecture Notes in Computer Science  
The running 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.  ... 
doi:10.1007/3-540-63397-9_5 fatcat:eboscpm7oze7xemzmltyzznswe

Solving polynomials by radicals with roots of unity in minimum depth

Gwoboa Horng, Ming-Deh Huang
1999 Mathematics of Computation  
Let α be a root of a 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.  ... 
doi:10.1090/s0025-5718-99-01060-1 fatcat:ffimh4x33necji62u24pcgvpfm

Limitations to algorithm solvability: Galois methods and models of computation

Chanderjit Bajaj
1986 Proceedings of the fifth ACM symposium on Symbolic and algebraic computation - SYMSAC '86  
x», where Q 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  ... 
doi:10.1145/32439.32453 dblp:conf/issac/Bajaj86 fatcat:okyzs77xyndizicj3276dcbm4a

The algebraic degree of geometric optimization problems

Chanderjit Bajaj
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.  ... 
doi:10.1007/bf02187906 fatcat:lkqtft64abfsheyntrzbf2u6na

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

Catherine DeGrandpre
2005 The Mathematics Enthusiast  
Since S 5 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.  ... 
doi:10.54870/1551-3440.1020 fatcat:gagf5urc7zbqtekdoojfhkfzve

A Complete Invariant Generation Approach for P-solvable Loops [chapter]

Laura Kovács
2010 Lecture Notes in Computer Science  
Our experimental results report that our method takes less iterations and/or 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.  ... 
doi:10.1007/978-3-642-11486-1_21 fatcat:swfitcpnhnervkr6e6apuws56q

Seismic Solvability Problems [article]

August Lau, Chuan Yin
2012 arXiv   pre-print
We can classify 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 (  ... 
arXiv:1212.1350v1 fatcat:ithynhp6obaxhop4a2orlo77zm

A Note on the Unsolvability of the Weighted Region Shortest Path Problem [article]

Jean-Lou De Carufel, Carsten Grimm, Anil Maheshwari, Megan Owen, Michiel Smid
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  ... 
arXiv:1305.5209v1 fatcat:zvhgnuk7pvhsrb6pdulpybqcqe

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

Andrew Walcott Beckwith
2019 Journal of High Energy Physics Gravitation and Cosmology  
how the physics answers are all 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] .  ... 
doi:10.4236/jhepgc.2019.51014 fatcat:ffoq7ktwxnh7xbn74di74wqusu

THE COMPLEXITY OF THE EQUIVALENCE PROBLEM OVER FINITE RINGS

GÁBOR HORVÁTH
2011 Glasgow Mathematical Journal  
We prove that for a finite ring if the factor 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.  ... 
doi:10.1017/s001708951100053x fatcat:wsaerkkpjndjffq35h543ylfla

Polynomial time algorithms for sentences over number fields

Shih Ping Tung
1992 Information and Computation  
We also show that ther are 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) .  ... 
doi:10.1016/0890-5401(92)90037-g fatcat:zzycxn2c6rbf3e6mky7ttczhqy

Practical approach to solvability: Geophysical application using complex decomposition into simple part (solvable) and complex part (interpretable) for seismic imaging [article]

August Lau, Chuan Yin
2010 arXiv   pre-print
The simple part 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.  ... 
arXiv:1012.0520v1 fatcat:e37keih2h5aezcps6zoi5rwnci

Upper Bounds on the Complexity of Some Galois Theory Problems [chapter]

V. Arvind, Piyush P Kurur
2003 Lecture Notes in Computer Science  
For 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.  ... 
doi:10.1007/978-3-540-24587-2_73 fatcat:kt3yapvwqfej7ay7mt5kk6dig4

Radically solvable graphs [article]

Bill Jackson, J. C. Owen
2012 arXiv   pre-print
It 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.  ... 
arXiv:1207.1580v1 fatcat:djwobc5dfrhj7argnqiody2mbm
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