On solving general algebraic equations by integrals of elementary functions

Mikhalkin, E. N.

doi:10.1007/s11202-006-0043-4

Cited by 8 publications

(6 citation statements)

References 2 publications

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“…Theorem 1 [6]. We can express the branch y 0 (x) of the solution to (1) as the integral y 0 (x) = 1 + 1 2πin…”

Section: The Power Series and Integral Representation For Y 0 (X)mentioning

confidence: 99%

“…Namely, a solution to (1) was obtained [6] as an integral over the segment. The convergence domain of this integral amounts to the complement to a complex ruled surface: it is neither a polydisk nor a polyhedral domain.…”

Section: Introductionmentioning

confidence: 99%

“…The convergence domain of this integral amounts to the complement to a complex ruled surface: it is neither a polydisk nor a polyhedral domain. Basing on the derived integral representation, [6] describes the monodromy of the solution y(x) to (1) with one parameter; i.e., in case only the variable x j in (1) is nonzero. To elucidate the gist of this study, observe that the discriminant set of (1) admits an n-valued HornKapranov parametrization x = Ψ(s) [3] (see Eq.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of Theorem 3 rests on a construction of two cuts Σ + and Σ − as real hypersurfaces obtained from the integral representation [6] for y 0 (x) (for the precise statement, see Theorem 1 below). Section 2 describes the differential geometry of these cuts.…”

Section: Introductionmentioning

confidence: 99%

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The monodromy of a general algebraic function

Mikhalkin

2015

Sib Math J

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We consider a general reduced algebraic equation of degree n with complex coefficients. The solution to this equation, a multifunction, is called a general algebraic function. In the coefficient space we consider the discriminant set ∇ of the equation and choose in its complement the maximal polydisk domain D containing the origin. We describe the monodromy of the general algebraic function in a neighborhood of D. In particular, we prove that ∇ intersects the boundary ∂D along n real algebraic surfaces S (j) of dimension n − 2. Furthermore, every branch y j (x) of the general algebraic function ramifies in D only along the pair of surfaces S (j) and S (j−1) .

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“…Theorem 1 [6]. We can express the branch y 0 (x) of the solution to (1) as the integral y 0 (x) = 1 + 1 2πin…”

Section: The Power Series and Integral Representation For Y 0 (X)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The monodromy of a general algebraic function

Mikhalkin

2015

Sib Math J

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“…The so-called Mellin integral formula [5] is the best known analytical algorithm for solving algebraic equations. Recently the further development of the Mellin integral formula was proposed [6].…”

Section: Introductionmentioning

confidence: 99%

The Solution of Algebraic Equations of Continuous Fractions of Nikiports

Shmoylov¹,

Kirichenko²,

Шмойлов³

et al. 2014

J. Sib. Fed. Univ., Math. phys.

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Analytical expressions representing all the roots of a random algebraic equation of n-th degree in terms of the equation coefficients are presented in the paper. These formulas consist of two ratios of infinite Toeplitz determinants. The diagonal elements of the determinants are the coefficients of algebraic equations. To find complex roots the method of summation of divergent continued fractions is used.Keywords: algebraic equation, infinite Toeplitz determinant, r/ϕ-algorithm, diverging continuous fractions. IntroductionTo design modern complex objects it is necessary to analyze their supposed characteristics even in the early stages of their development. Differential and integral equations as well as the classical algebraic equations are used for modeling. Algebraic equation is one of the oldest objects of research in mathematics.There are various applications of algebraic equations to scientific and technical problems. For example, algebraic equations arise is of interest in studies of equilibrium states of complex thermodynamic and mechanical systems. Algebraic equations are often used in aerodynamics. For example, the rate of climb of a plane is determined by the algebraic equation of eighth degree. Algebraic equations are used in the calculation of the flow over the wing in the Prandtl theory. The degree of the equations depends on the law of variation of the lift coefficient from the angle of attack. The problem of structural stability involves the calculation of eigenvalues of matrices. The eigenvalues are determined from the solution of algebraic equation. The equation degree is equal to the number of harmonics. Algebraic equations most often arise in various geometrical calculations, for example, in determination of the intersection points of two curves, in the design of wings, fuselages, etc. The recently published monograph [1,2] dedicated to the various aspects of the theory and practice of algebraic equations. However, internationally known expert R. Hamming gave, in the book published half a century ago [3], the following remark: "the problem of finding the roots of polynomials occurs frequently enough to warrant a thorough examination and development of special methods for its solution. A whole book can be dedicated to the different known methods of finding a valid linear and quadratic multiplier. The large verity of methods shows that no one of them is completely satisfactory". In fact, there * shmoylov40@atinfotectt.ru † vtgak@mail.ru c Siberian Federal University. All rights reserved -533 - The paper presents analytical expressions that represent all the roots of a random algebraic equation of n-th degree in terms of the coefficients of the equation. These formulas consist of two ratios of infinite Toeplitz determinants. The diagonal elements of the determinants are the coefficients of algebraic equations. To find complex roots an advanced method of summation of divergent continued fractions, called r/ϕ-algorithm, is used [7]. The algorithm is finding application in various areas of computati...

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The uniformization of certain algebraic hypergeometric functions

Maier

2014

Advances in Mathematics

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The hypergeometric functions n F n−1 are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic n F n−1 's, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this so-called Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic n F n−1 's. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit single-valued or multivalued parametrizations of individual algebraic n F n−1 's, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.

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On solving general algebraic equations by integrals of elementary functions

Cited by 8 publications

References 2 publications

The monodromy of a general algebraic function

The monodromy of a general algebraic function

The Solution of Algebraic Equations of Continuous Fractions of Nikiports

The uniformization of certain algebraic hypergeometric functions

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