Solvable Polynomial Research Articles

Solvable Polynomial Ideals: The Ideal Reflection for Program Analysis

This paper presents a program analysis method that generates program summaries involving polynomial arithmetic. Our approach builds on prior techniques that use solvable polynomial maps for summarizing loops. These techniques are able to generate all polynomial invariants for a restricted class of programs, but cannot be applied to programs outside of this class---for instance, programs with nested loops, conditional branching, unstructured control flow, etc. There currently lacks approaches to apply these prior methods to the case of general programs. This paper bridges that gap. Instead of restricting the kinds of programs we can handle, our method abstracts every loop into a model that can be solved with prior techniques, bringing to bear prior work on solvable polynomial maps to general programs. While no method can generate all polynomial invariants for arbitrary programs, our method establishes its merit through a monotonicty result. We have implemented our techniques, and tested them on a suite of benchmarks from the literature. Our experiments indicate our techniques show promise on challenging verification tasks requiring non-linear reasoning.

Fourteen years of cube attacks

AbstractAlgebraic Cryptanalysis is a widely used technique that tackles the problem of breaking ciphers mainly relying on the ability to express a cryptosystem as a solvable polynomial system. Each output bit/word can be expressed as a polynomial equation in the cipher’s inputs—namely the key and the plaintext or the initialisation vector bits/words. A part of research in this area consists in finding suitable algebraic structures where polynomial systems can be effectively solved, e.g., by computing Gröbner bases. In 2009, Dinur and Shamir proposed the cube attack, a chosen plaintext algebraic cryptanalysis technique for the offline acquisition of an equivalent system by means of monomial reduction; interpolation on cubes in the space of variables enables retrieving a linear polynomial system, hence making it exploitable in the online phase to recover the secret key. Since its introduction, this attack has received both many criticisms and endorsements from the crypto community; this work aims at providing, under a unified notation, a complete state-of-the-art review of recent developments by categorising contributions in five classes. We conclude the work with an in-depth description of the kite attack framework, a cipher-independent tool that implements cube attacks on GPUs. Mickey2.0 is adopted as a showcase.

Irreducible Radical Field Extensions

This paper is devoted to irreducible radical field extensions, i.e., extensions that can be obtained by adjunction of roots of irreducible binomials. We find a criterion for cyclotomic extensions to be irreducible radical. This criterion developes the Gauss–Wantzel theorem about constructible polygons. We also prove that any normal solvable extension of some field K is irreducible radical until K has all roots of unity. This generalizes Abel’s theorem, which fills the gap in the uncomplete Ruffini proof of the impossibility theorem for the general equation of degree five or higher. Finally, we prove that the root set of any irreducible solvable polynomial coincides with the value set of some radical formula using irreducible radical extensions.

On the algebras : from a constructive-computational viewpoint

Let (resp. ) be the -part (resp. -part) of the Drinfeld-Jimbo quantum group of type AN over a field K. With respect to Jimbo relations and the PBW K-basis of (resp. ) established by Yamane, it is shown, by constructing an appropriate monomial ordering on , that (resp. ) is a solvable polynomial algebra. Consequently, further structural properties of (resp. ) and their modules may be established and realized in a constructive-computational way.

Generalized down-up algebras revisited from a viewpoint of Gröbner basis theory

The so called generalized down-up algebras are revisited from a viewpoint of Gröbner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided ), and by means of homogeneous Gröbner defining relations, the associated graded structures of generalized down-up algebras, namely the associated graded algebras, Rees algebras, and the homogenized algebras of generalized down-up algebras, are explored comprehensively.

Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings

This work's primary objective is to investigate an effective Buchberger-Zacharias-Weispfenning theory of bimodules over skew polynomial extensions of a coefficient ring satisfying the appropriate notion of coherence. The original contribution of our approach establishes the subbilateral coherence together with the effective computation of subbilateral Gröbner bases for a sufficiently large class of skew polynomial extensions, over term-ordered monoids, of subbilateral Noetherian coherent coefficient rings. The strategy to produce finite bilateral Gröbner bases from subbilateral ones generalizes Kandri-Rody and Weispfenning completion from left Gröbner bases to bilateral ones over a subclass of solvable polynomial rings, as well as Weispfenning completion from Weispfenning restricted modules to bilateral ones over the bivariate Weispfenning extensions of skew fields. It equally results that the class of rings that are both left Noetherian and left coherent (or resp., right Noetherian and right coherent) is closed under iterative skew polynomial extension with bijective conjugation maps.

Quasi-exactly solvable polynomial extensions of the quantum harmonic oscillator

The (analytic) sextic oscillator is often considered as the prototype of quasi-exactly solvable (QES) Schrödinger equations, i.e., those Schrödinger equations for which at some ad hoc couplings a finite number of eigenstates can be found explicitly by algebraic means, while the remaining ones remain unknown. Recently, a (non-analytic) QES symmetrized quartic oscillator was introduced and shown to complete the list of (analytic) QES anharmonic oscillators, which does not contain any quartic one. Here we prove that such a quartic oscillator is amenable to an sl(2, ℝ) description. Furthermore, we generalize it to a symmetrized sextic oscillator. The latter is obtained by parting the real line into two subintervals ℝ− and ℝ+ on which the corresponding Schrödinger equations are solved by using the functional Bethe ansatz method, and the resulting wavefunctions and their first derivatives are matched at x = 0. Two categories of QES potentials are obtained: the first one containing the well-known analytic sextic potentials as a subset, and the second one of novel potentials with no counterpart in such a class.

An elimination lemma for algebras with PBW bases

ABSTRACTLet K be a field, and a finitely generated K-algebra with the PBW K-basis . It is shown that if L is a nonzero left ideal of A with GK.dim(A∕L) = d<n ( = the number of generators of A), then L has the elimination property in the sense that V(U)∩L≠{0} for every subset with , where V(U) = K-span. In terms of the structural properties of A, it is also explored when the condition GK.dim(A∕L)<n may hold for a left ideal L of A. Moreover, from the viewpoint of realizing the elimination property by means of Gröbner bases, it is demonstrated that if A is in the class of binomial skew polynomial rings or in the class of solvable polynomial algebras, then every nonzero left ideal L of A satisfies GK.dim(A∕L)< GK.dimA = n ( = the number of generators of A), thereby L has the elimination property.

On computation of minimal free resolutions over solvable polynomial algebras

Let $A=K[a_1,\ldots ,a_n]$ be a (noncommutative) solvable polynomial algebra over a field $K$ in the sense of A. Kandri-Rody and V. Weispfenning [Non-commutative Grobner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), 1--26]. This paper presents a comprehensive study on the computation of minimal free resolutions of modules over $A$ in the following two cases: (1) $A=\bigoplus_{p\in\mathbb{N}}A_p$ is an $\mathbb{N}$-graded algebra with the degree-0 homogeneous part $A_0=K$; (2) $A$ is an $\mathbb{N}$-filtered algebra with the filtration $\{F_pA\}_{p\in\mathbb{N}}$ determined by a positive-degree function on $A$.

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings

Abstract Signature-based algorithms are efficient algorithms for computing Gröbner-Shirshov bases in commutative polynomial rings, and some noncommutative rings. In this paper, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings over fields.

Roots of solvable polynomials of prime degree

An explicit formula for the most general root of a solvable polynomial of prime degree is stated and proved. Such a root can be expressed rationally in terms of a single compound radical determined by the roots of a cyclic polynomial whose degree divides μ−1, where μ is the prime. The study of such formulas was initiated by a formula of Abel for roots of quintic polynomials that are solvable, and was carried forward by Kronecker and a few others, but seems to have lain dormant since 1924. A formula equivalent to the one given here is contained in a paper of Anders Wiman in 1903, but it seems to have been forgotten.

A Weak Approximation of Stochastic Differential Equations with Jumps Through Tempered Polynomial Optimization

We present an optimization approach to the weak approximation of a general class of stochastic differential equations with jumps, in particular, when value functions with compact support are considered. Our approach employs a mathematical programming technique yielding upper and lower bounds of the expectation, without Monte Carlo sample paths simulations, based upon the exponential tempering of bounding polynomial functions to avoid their explosion at infinity. The resulting tempered polynomial optimization problems can be transformed into a solvable polynomial programming after a minor approximation. The exponential tempering widens the class of stochastic differential equations for which our methodology is well defined. The analysis is supported by numerical results on the tail probability of a stable subordinator and the survival probability of Ornstein-Uhlenbeck processes driven by a stable subordinator, both of which can be formulated with value functions with compact support and are not applicable in our framework without exponential tempering.

The construction of solvable polynomials

The construction of solvable polynomials

Errata to ‘‘The construction of solvable polynomials”

in order to include the case ν = 1. (When ν = 1, the argument of footnote 12 fails and no δ satisfies the requirements of the theorem. When the formula is stated as above, no δ is called for when ν = 1.) In the special case ν = 1, the definition of G(x) in (5.2) should be replaced by G(x) = x − sμ0 , where s0 is a nonzero Lagrange resolvent. (In this case, there is no m, but this G(x) has the μ needed roots αs0.) When ν = 1, the assertion to be proved in Section 7 reduces to a tautology. 2. Proposition 4.1 contains a serious error that does not affect the rest of the paper. The formula αsi → αsi+κ it gives for τ describes a permutation of the Lagrange resolvents, but does not describe an automorphism of Ω, so the proposition fails to provide the needed τ . (The τ constructed in the proof is an automorphism, but it does not combine with σ and η to generate the group.) Correction of the formula for τ implies corrections in the relations, but the main assertions remain:

Solvability by Real Radicals and Fermat Primes

AbstractWe give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

On the number of real roots of a solvable polynomial

1. Introduction and Loewy’s theorem. By a classical theorem the number of real roots of an irreducible polynomial f(X) of odd prime degree p over a real number eld K is either 1 or p if the Galois group of f(X) over K is solvable. This result was generalized by A. Loewy in the following way: For a polynomial f(X) we let r(f) denote the number of real roots of f(X). Loewy’s theorem. Let K be a real number eld and f(X) an irreducible polynomial in K[X] of odd degree n. If p is the smallest prime divisor of n and the Galois group of f(X) over K is solvable, then r(f) = 1 or n or satises the inequalities p r(f) n p + 1. When the degree of f(X) is a prime number the above theorem is an immediate corollary to the following Galois’ theorem. Let f(X) be an irreducible separable polynomial over a eld K having a solvable Galois group over K. If the degree of f(X) is a prime number, then any two roots of f(X) generate the splitting eld of f(X) over K. Galois’ theorem, which is basically a group-theoretic result, cannot be generalized to yield a proof of Loewy’s theorem. Indeed, for any odd prime number p and any t, 1 t p, there exists an irreducible polynomial f(X) in Q[X] of degree p 2 with solvable Galois group having t roots 1;:::; t

Regularity and K0-group of quadric solvable polynomial algebras

Concerning solvable polynomial algebras in the sense of Kandri-Rody and Weispfenning [J. Symbolic Comput. 9 (1990) 1–26], it is shown how to recognize and construct quadric solvable polynomial algebras in an algorithmic way. If A= k[ a 1,…, a n ] is a quadric solvable polynomial algebra, it is proved that gl.dim A⩽ n and K 0(A)≅ Z . If A is a tame quadric solvable polynomial algebra, it is shown that A is completely constructable and Auslander regular.

Solvable Brioschi Resolvents

The Brioschi resolvent y 5 −10Zy 3 +45Z 2 y−Z 2 is a key component of an algorithm for calculating the roots of a general quintic polynomial. It is obtained from the general quintic polynomial by applying two Tschirnhausen transformations. In this paper it is shown that if the quintic polynomial is a solvable polynomial, then its associated parameter Z in the Brioschi resolvent satisfies Z=g(t) where g(t) is a rational function in ℚ(t) and t is chosen from an appropriate field.

Reduction of Permutation-Invariant Polynomials

Let R be a commutative ring with 1, let R 〈 X 1 ,…, X n 〉 / I be the polynomial algebra in the n ≥4 noncommuting variables X 1 ,…, X n over R modulo the set of commutator relations I ={( X 1 +···+ X n )∗ X i = X i ∗( X 1 +···+ X n )|1≤ i ≤ n }. Furthermore, let G be an arbitrary group of permutations operating on the indeterminates X 1 ,…, X n , and let R 〈 X 1 ,…, X n 〉 / I G be the R -algebra of G -invariant polynomials in R 〈 X 1 ,…, X n 〉 / I . The first part of this paper is about an algorithm, which computes a representation for any f ∈ R 〈 X 1 ,…, X n 〉 / I G as a polynomial in multilinear G -invariant polynomials, i.e., the maximal variable degree of the generators of R 〈 X 1 ,…, X n 〉 / I G is at most 1. The algorithm works for any ring R and for any permutation group G . In addition, we present a bound for the number of necessary generators for the representation of all G -invariant polynomials in R 〈 X 1 ,…, X n 〉 / I G with a total degree of at most d . The second part contains a first but promising analysis of G -invariant polynomials of solvable polynomial rings.