Особенности дискриминантных множеств систем полиномов Лорана

Let $X$ be the family of hypersurfaces in the odd-dimensional torus ${\mathbb T}^{2n+1}$ defined by a Laurent polynomial $f$ with fixed exponents and variable coefficients. We show that if $nΔ$, the dilation of the Newton polytope $Δ$ of $f$ by the factor $n$, contains no interior lattice points, then the Picard-Fuchs equation of $W_{2n}H^{2n}_{\rm DR}(X)$ has a full set of algebraic solutions (where $W_\bullet$ denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].

We study the distribution of Mahler′s measures of reciprocal polynomials with complex coefficients and bounded even degree. We discover that the distribution function associated to Mahler′s measure restricted to monic reciprocal polynomials is a reciprocal (or antireciprocal) Laurent polynomial on [1, ∞) and identically zero on [0, 1). Moreover, the coefficients of this Laurent polynomial are rational numbers times a power of π. We are led to this discovery by the computation of the Mellin transform of the distribution function. This Mellin transform is an even (or odd) rational function with poles at small integers and residues that are rational numbers times a power of π. We also use this Mellin transform to show that the volume of the set of reciprocal polynomials with complex coefficients, bounded degree, and Mahler′s measure less than or equal to one is a rational number times a power of π.

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry.

We propose a new approach (through Fourier decomposition of currents on the complex torus) to the notion of Ronkin function $$R_f$$ of a Laurent polynomial with complex coefficients and extend it to that of Ronkin current $$R_{f}$$ , where f is more generally a Laurent polynomial mapping $$(f_1,\ldots ,f_m): \mathbb {T}^n \rightarrow \mathbb {C}^m$$ . The concept of Ronkin function was introduced by Ronkin (On zeroes of almost periodic functions generated by holomorphic functions in a multicircular domain, Complex analysis in modern mathematics (in Russian), Fazis, Moscow, pp 243–256, 2000); it appears to be closely related to that of archimedean amœba introduced by Gelfand et al. (Discriminants, resultants and multidimensional determinants, mathematics: theory and applications, Birkhauser Boston, Inc., Boston, 1994); the interest of such notions in pure or applied mathematics lie in the fact they connect complex geometry with max-plus (so called tropical) geometry. Inspired precisely by image processing methods in the particular case $$n=2$$ , we use the Laplace differential operator (acting on distributions) in order to visualize the contour of the corresponding archimedean amœba $${\mathscr {A}}_f$$ . We also introduce a notion of refined spine in accordance with the persistence of the geometric genus under complex versus tropical deformation of $${\mathscr {A}}_f$$ . Numerical illustrations conducted under Sage and Matlab (with codes provided) are detailed and commented as illustrations. Ronkin function as well as Ronkin current are also interpreted from the pluripotential point of view as Green currents with respect to the affine or toric Lelong-Poincare equation, the setting (here toric) being inspired by that of the $$(d',d'')$$ -differential calculus on $$\mathbb {R}^n$$ (based on the introduction of a “ghost” copy of $$\mathbb {R}^n$$ ) such as introduced by Lagerberg (Math Z 270(3–4):1011–1050, 2012). Related potential applications to diophantine geometry, following a recent result by Gualdi (Heights of hypersurfaces in toric varieties, preprint, arXiv:1711.00710 , 2017), are finally discussed.

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials, are also sparse. The proofs are based on a generalization of a toric Bertini's theorem due to Fuchs, Mantova and Zannier.

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a fewnomial hypersurface in R N. In the book Fewnomials (6), A. Khovanskii gives bounds on the Betti numbers of varieties X in R n or in the positive orthant R n . These varieties include algebraic varieties, varieties defined by polynomial functions in the variables and exponentials of the variables, and also by even more general functions. Here, we consider only algebraic varieties. For these, Khovanskii bounds the sum b∗(X) of Betti numbers of X by a function that depends on n, the codimension of X, and the total number of monomials appearing (with non zero coefficients) in polynomial equations definingX. For real algebraic varieties X, the problem of bounding b∗(X) has a long history. O. A. Oleinik (8) and J. Milnor (7) used Morse theory to estimate the number of critical points of a suitable Morse function to obtain a bound. R. Thom (12) used Smith Theory to bound the mod-2 Betti numbers. This Smith-Thom bound has the form b∗(X) ≤ b∗(XC), where XC is the complex variety defined by the same polynomials as X. For instance, if X ⊂ R n is defined by polynomials of degree at most d, then Milnor's bound is b∗(X) ≤ d(2d − 1) n . If X ⊂ (R {0}) n is a smooth hypersurface defined by a polynomial with Newton polytope P , then the Smith-Thom bound is b∗(X) ≤ b∗(XC) = n! ¢ Vol(P ), where Vol(P ) is the usual volume of P. We refer to (10) for an informative history of the subject, and to the book (1) for more details. These bounds are given in term of degree or volume of a Newton polytope which are numerical deformation invariants of the complex variety XC. In contrast, the topology of a real algebraic variety depends on the coefficients of its defini ng equations, and in partic- ular, on the number of monomials involved in these equations. For instance, Descartes's rule of signs implies that the number of positive roots of a real univariate polynomial is less than its number of monomials, but the number of complex roots is equal to its degree. Khovanskii's bound can be seen as a generalization of this Descartes bound. It is smaller than the previous bounds when the defining equations have few monomials compared to their degrees. While it has always been clear that Khovanskii's bounds are unrealistically large, it appears very challenging to sharpen them. Some progress has been made recently for the number of non degenerate positive solutions to a system of n polynomials in n variables (3). Here, we bound b∗(X), when X is a fewnomial hypersurface. Suppose that X ⊂ R n is a smooth hypersurface defined by a Laurent polynomial with

<!-- *** Custom HTML *** --> The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential function, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of doubly stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors.

Comment on “On the numerical simulation of particle dynamics in the radiation belt. Part I: Implicit and semi‐implicit schemes” and “On the numerical simulation of particle dynamics in the radiation belt. Part II: Procedure based on the diagonalization of the diffusion tensor” by E. Camporeale et al.

The paper discusses options for the identification of the neighborhood system, in cases where there is only one set of parameter values - the nominal mode; a set of tuples of state and control data; range of valid parameter values. A procedure of mixed control with variable coefficients is proposed with the introduction of a fuzzy coefficient, which characterizes the relationship between the nodes of the system and shows the possible appearance of risks in the system.

The aim of this research was to determine the optimal dimensions of a tapered, bisymmetric, thin-walled, I-shaped cantilever for which, at a constant mass, the critical load causing the buckling of the beam reaches the maximum value. Optimal solutions were sought in a discrete set of admissible values of the taper coefficients of the I-bar’s web and/or flanges. A model of a thin-walled bar with an open cross-section, whose mathematical description was derived on the basis of the momentless theory of shells, was used to analyse the considered problem described by a system of four coupled differential equations with variable coefficients. The equations were solved using Chebyshev series of the first kind to approximate the generalized displacement functions, and the recurrence algorithm presented in earlier publications by the author(s). A tapered cantilever with a bisymmetric cross-section, subjected to the action of a uniformly distributed load, was analysed. The load can be applied to the I-bar’s upper flange or to its lower flange, or it can act in the centre of the web. The obtained critical load values were compared with those obtained using the finite element method and the commercial Sofistik software. In the set of linearly tapered cantilevers with a bisymmetric double-tee cross section, the cantilevers with the highest taper coefficients of the web and the flanges (the free end having the smallest possible dimensions) were found to be optimal (in their case, at a constant mass, the critical buckling load reaches its maximum value).

This study investigates the theoretical and practical aspects of fourth-order elastic beam equations characterized by variable coefficients and influenced by combined nonlinearities. The equations under consideration arise naturally in the context of high-order differential problems encountered in material science, particularly in the study of phase transitions. The research focuses on establishing the existence and multiplicity of solutions for these equations, which involve both concave and convex nonlinearities. Using advanced mathematical techniques, such as variational methods and critical point theory, the study provides rigorous proofs for the existence of solutions under specific conditions. A central result is the application of Bonanno's local minimum theorem, which ensures the existence of at least one solution. Moreover, the research shows that by imposing additional algebraic conditions, particularly the classical Ambrosetti–Rabinowitz condition, the presence of two distinct solutions can be guaranteed. Beyond this, critical point theorems from Averna and Bonanno are employed to demonstrate scenarios where three solutions are possible.

Like the interval model of Kharitonov, the diamond model proves to be an alternative powerful device for taking into account the variation of parameters in prescribed ranges. The robust stability of some kinds of diamond polynomial families with complex coefficients are discussed. By exploiting the geometric characterizations of their value sets, we show that, for the family of polynomials with complex coefficients and both their real and imaginary parts lying in a diamond, the stability of eight specially selected extreme point polynomials is necessary as well as sufficient for the stability of the whole family. For the so-called simplex family of polynomials, four extreme point and four exposed edge polynomials of this family need to be checked for the stability of the entire family. The relations between the stability of various diamonds are also discussed.

Application of biomechanical models relies on model parameters estimated from experimental data. Parameter non-identifiability, when the same model output can be produced by many sets of parameter values, introduces severe errors yet has received relatively little attention in biomechanics and is subtle enough to remain unnoticed in the absence of deliberate verification. The present work develops a global identifiability analysis method in which cluster analysis and singular value decomposition are applied to vectors of parameter-output variable correlation coefficients. This method provides a visual representation of which specific experimental design elements are beneficial or harmful in terms of parameter identifiability, supporting the correction of deficiencies in the test protocol prior to testing physical specimens. The method was applied to a representative nonlinear biphasic model for cartilaginous tissue, demonstrating that confined compression data does not provide identifiability for the biphasic model parameters. This result was confirmed by two independent analyses: local analysis of the Hessian of a sum-of-squares error cost function and observation of the behaviour of two optimization algorithms. Therefore, confined compression data are insufficient for the calibration of general-purpose biphasic models. Identifiability analysis by these or other methods is strongly recommended when planning future experiments.

Lasso Penalty method for variable selection in database construction process and developing house value models in RUA