In this paper, an algorithm for calculating the parameterization of the Horn–Kapranov A -discriminant set and singular points of an algebraic hypersurface, using the Maple computer algebra system, is proposed.

We consider discriminant Δ _n of a general polynomial of degree n and Newton polytope 𝒩 for the discriminant, and prove that Newton polytopes with truncations of Δ _n with respect to faces 𝒩 are the Minkowski sums of Newton polytopes for the discriminants of the polynomials of lower degrees.

A program that computes the truncation of the discriminant for a polynomial of one variable onto facets of the Newton polytope for the discriminant of this polynomial, as well as enables the factorization of this truncation into the product of discriminants of polynomials of lower degrees, is developed.

A general polynomial in one variable is considered and the explicit factorization formulas for the truncations of the discriminant with respect to coordinate faces of the polynomial Newton polytope are presented. As a result, the extension of the formulas presented by Gelfand–Kapranov–Zelevinsky is obtained

. In modern mathematics by development of algorithmic and computer methods formulas for solving polynomial equations are considered in more details. In the paper a polynomial equation of fourth power with one parameter is considered. Such equations are called trinomial. For it such methods of solution are known as methods of Ferrari, Descartes and Euler. An approach is used based on Mellin and Belardinelli integral representations, and also usage of the inverse Mellin transform. As a main result the formula is proved for solutions obtained by Euler – Descartes method with representations of series for hypergeometric functions.

The Horn–Kapranov parametrizations describe the singular sets of hypergeometric functions in several variables. These parametrizations are inverses of logarithmic Gauss maps for A-discriminants. In this paper we demonstrate that, despite the multivalued nature of the indicated parametrizations, their blow-ups properties are the same as for single-valued meromorphic mappings. As an application, a new proof of factorization identities for the classical discriminant is given.

Let Δn be the discriminant of a general polynomial of degree n and $$\mathcal{N}$$ be the Newton polytope of Δn. We give a geometric proof of the fact that the truncations of Δn to faces of $$\mathcal{N}$$ are equal to products of discriminants of lesser n degrees. The proof is based on the blow-up property of the logarithmic Gauss map for the zero set of Δn.

Рассматривается общий многочлен с переменными коэффициентами. В терминах результантов многочлена и его производных получены простые рациональные выражения от коэффициентов для кратных корней многочлена. Аналогичные результаты распространяются для систем $n$ полиномиальных уравнений от $n$ неизвестных. Обоснования полученных формул кратных корней базируются на использовании свойств логарифмического отображения Гаусса для дискриминантного множества системы уравнений и процедуры линеаризации системы. Полученные формулы представляют интерес не только в теоретическом аспекте алгебры многочленов, но также в вычислительной математике и в различных разделах прикладной математики, связанных с нахождением критических точек полиномиальных отображений. Библиография: 20 названий.

We consider a complex hypersurface V given by an algebraic equation in k unknowns, where the set A subset of Z(k) of monomial exponents is fixed, and all the coefficients are variable. In other words, we consider a family of hypersurfaces in (C \ 0)(k) parametrized by its coefficients a = (a(alpha)) (alpha is an element of A) is an element of C-A. We prove that when A generates the lattice Z(k) as a group, then over the set of regular points a in the A-discriminantal set, the singular points of V admit a rational expression in a.

The general polynomial with variable coefficients is considered. In terms of the resultants of this polynomials and its derivatives simple rational expressions in the coefficients of the polynomial are found for its multiple zeros. Similar results are extended to systems of polynomial equations with unknowns. Justifications of the formulae for multiple roots thus obtained are based on the properties of the logarithmic Gauss map of the discriminant variety of a system of equations and on a linearization procedure for the system. The resulting formulae are of interest not only for theoretical aspects of the algebra of polynomials, but also for numerical mathematics and various areas of applied mathematics connected with finding critical points of polynomial maps. Bibliography: 20 titles.

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 \(S^{(j - 1)} \).

We consider an algebraic equation with variable complex coefficients. For the reduced discriminant set of such an equation we obtain parametrizations of the singular strata corresponding to the existence of roots of multiplicity at least j. These parametrizations are the restrictions of the Horn-Kapranov parametrization of the whole discriminant set to a chain of nested linear subspaces of the projective space. It is proved that such strata can be transformed into reduced A-discriminant sets by monomial transformations.

Рассматривается алгебраическое уравнение с переменными комплексными коэффициентами. Для приведенного дискриминантного множества такого уравнения получены параметризации сингулярных стратов, отвечающих за наличие корней кратности не меньше $j$. Эти параметризации являются сужениями параметризации Горна-Капранова всего дискриминантного множества на цепочку вложенных линейных подпространств проективного пространства. Доказано, что такие страты мономиальными преобразованиями переводятся в приведенные $A$-дискриминантные множества. Библиография: 12 названий.

Consider a general polynomial of degree n with variable coefficients. It is known that the Newton polytope of its discriminant is combinatorially equivalent to an (n-1)-dimensional cube. We show that two facets of this Newton polytope are prisms, and that truncations of the discriminant with respect to facets factor into discriminants of polynomials of smaller degree.

An amoeba of an analytic set is the real part of its image in a logarithmic scale. Among all hypersurfaces A-discriminantal sets have the most simple amoebas. We prove that any singular cuspidal stratum of the classical discriminant can be transformed by a monomial change of variables into an A-discriminantal set and compute the contours of the amoebas of these strata.

Algebraic equations with one and two parameters are considered. We prove that solutions to such equations can be represented as linear combination of generalized hypergeometric series. This result allows to express (nonlinearly) solutions to cubic and quartic equations by Gauss hypergeometric series.

A complete list of power series (centered at the point x = 0) is obtained for the solution y(x) of the general reduced algebraic equation \(y^n x_s y^{n_s } + ... + x_1 y^{n_1 } - 1 = 0\). The domains of convergence of these series are described in terms of the amoeba of the discriminant of the equation. Sectorial domains through which one selected series is analytically continued to the other series are explicitly described.

In this paper we establish a relationship between two approaches to the solution of algebraic fifth-degree equations, namely, the Hermite-Kronecker method (based on the modular elliptic equation) and the Mellin method (based on hypergeometric series).