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The Turán problem asks for the largest number of edges ex(n, H) in an n-vertex graph not containing a fixed forbidden subgraph H, which is one of the most important problems in extremal graph theory. However, the order of magnitude of ex(n, H) for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomial resultant approach to the algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lower bounds on the Turán number of 1-subdivision of K 3 , t 1 and the linear Turán number of the Berge theta hypergraph Θ 3 , t 2 B , where t1 = 25 and t2 = 217. Moreover, the constant t1 improves the random algebraic construction of Bukh and Conlon (2018) and makes the known estimation better on the smallest value of t1 concerning a problem posed by Conlon et al. (2021) by reducing the value from a magnitude of 1056 to the number 25, while the constant t2 improves a result of He and Tait (2019). [ABSTRACT FROM AUTHOR]

Piecewise linear differential systems separated by two parallel straight lines of the type of centre-centre-Hamiltonian saddle and the centre-Hamiltonian saddle-Hamiltonian saddle can have at most one limit cycle and there are systems in these classes having one limit cycle. In this paper, we study the limit cycles of a piecewise smooth differential system separated by two parallel straight lines formed by nonlinear centres and a Hamiltonian saddle. [ABSTRACT FROM AUTHOR]

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main motivating example, we calculate the rational homology groups of spaces of even and odd maps S m → S M , m < M , or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ℝ m + 1 → ℝ M + 1 of increasing degree. Also, we calculate the homology groups of spaces of ℤ r -equivariant maps of odd-dimensional spheres for any r. In auxiliary calculations, we find the homology groups of configuration spaces of projective and lens spaces with coefficients in several local systems. [ABSTRACT FROM AUTHOR]

We consider systems of two equations consisting of a polynomial and an entire function. By calculating the resultant of the polynomial and the entire function in two different ways, we can obtain a relation for double numerical series. The formula of A. M. Kytmanov and E. K. Myshkina was used as the first method for calculating the resultant. For the second method, we chose the formula for product of values of one function in the roots of another. A family of sums of some types of double numerical series absent in known references was found. We also demonstrate an approach to finding sums of lower dimension (one-dimensional sums) that arise when calculating the resultant of the original system of functions.

The work is devoted to the study of the number of real roots of systems of transcendental equations in $\mathbb C^n$ with real coefficients, consisting of entire functions, in some bounded multidimensional domain $D\subset \mathbb R^n$. It is assumed that the number of roots of the system is discrete (then it is no more than countable). For some entire function $\varphi (z), z\in \mathbb C^n$, with real Taylor coefficients at $z=0$, and a given system of equations, the concept of a resultant $R_\varphi(t)$ is introduced, which is an entire function of one complex variable $t$. It is constructed using power sums of the roots of the system in a negative degree, found using residue integrals. If the resultant has no multiple zeros, then it is shown that the number of real roots of the system in $D=\{x\in \mathbb R^n: a

This work focuses on developing an analytical method for constructing images in gravitational lenses. Building on our previously proposed elimination method, which utilized algebraic geometry for constructing images from circular sources, this study emphasizes images from elliptical sources using axially symmetric gravitational lenses. We detail the rational parameterization of an arbitrarily located source and use our algorithm to derive an expression that determines all images for an arbitrary axially symmetric gravitational lens. To validate this expression, we considered several limiting cases leading to well-known results. Specifically, we examined two simple models: a single-point lens and a homogeneous disk lens. By placing an elliptical source at the origin of the source plane and reducing it to a circle, we reproduced all previously known images, such as the Einstein ring for a single-point lens, and a double Einstein ring for a disk lens. Additionally, we demonstrated the Construction of images in other arbitrary lens-source configurations. [ABSTRACT FROM AUTHOR]

It is evident that skew polynomials offer promising directions for developing cryptographic schemes. This paper focuses on exploring skew polynomials and studying their properties with the aim of exploring their potential applications in fields such as cryptography and combinatorics. We begin by deriving the concept of resultants for bivariate skew polynomials. Then, we employ the derived resultant to incrementally eliminate indeterminates in skew polynomial systems, utilising both direct and modular approaches. Finally, we discuss some applications of the derived resultant, including cryptographic schemes (such as Diffie–Hellman) and combinatorial identities (such as Pascal's identity). We start by considering a bivariate skew polynomial system with two indeterminates; our intention is to isolate and eliminate one of the indeterminates to reduce the system to a simpler form (that is, relying only on one indeterminate in this case). The methodology is composed of two main techniques; in the first technique, we apply our definition of a (bivariate) resultant via a Sylvester-style matrix directly from the polynomials' coefficients, while the second is based on modular methods where we compute the resultant by using evaluation and interpolation approaches. The idea of this second technique is that instead of computing the resultant directly from the coefficients, we propose to evaluate the polynomials at a set of valid points to compute its corresponding set of partial resultants first; then, we can deduce the original resultant by combining all these partial resultants using an interpolation technique by utilising a theorem we have established. [ABSTRACT FROM AUTHOR]

We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be written as conjunctions of several polynomial equalities and one inequality in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It is also known that discriminants can be obtained by using repeated parametric gcd's. The resulting discriminants are usually nested determinants, i.e., determinants of matrices whose entries are determinants, and so on. In this paper, we give a new type of discriminants which are not based on repeated gcd's. The new discriminants are simpler in the sense that they are non-nested determinants and have smaller maximum degrees. [ABSTRACT FROM AUTHOR]

We consider homogeneous binomial ideals I = (f 1 , ... , f n) in K [ x 1 , ... , x n ] , where f i = a i x i d i − b i m i and a i ≠ 0. When such an ideal is a complete intersection, we show that the monomials which are not divisible by x i d i for i = 1 , ... , n form a vector space basis for the corresponding quotient, and we describe the Macaulay dual generator in terms of a directed graph that we associate to I. These two properties can be seen as a natural generalization of well-known properties for monomial complete intersections. Moreover, we give a description of the radical of the resultant of I in terms of the directed graph. [ABSTRACT FROM AUTHOR]

Holditch’s theorem is a classical geometrical result on the areas of a given closed curve and another one, its Holditch curve, which is constructed as the locus of a fixed point dividing a chord of constant length that moves with its endpoints over the given curve and that returns back to its original position after some full revolution. Holditch curves have already been studied from the parametric point of view, although numerical methods and approximations are often necessary for their computation. In this paper, implicit equations of Holditch curves of algebraic curves are studied. The implicit equations can be simply found from the computation of a resultant of two polynomials. With the same techniques, Holditch curves of two initial algebraic curves are also considered. Moreover, the use of implicit equations allows to find new and explicit parameterizations of non-trivial Holditch curves, such as in the case of having an ellipse as an initial curve. [ABSTRACT FROM AUTHOR]

In this article, we extend the classical framework for computing discriminants of special quasi-orthogonal polynomials from Schur's resultant formula, and establish a framework for computing discriminants of a sufficiently broader class of polynomials from the resultant formulas that are proven by Ulas and Turaj. More precisely, we derive a formula for the discriminant of a sequence { r A , n + c r A , n - 1 } of polynomials. Here, c is an element of a field K and { r A , n } is a sequence of polynomials satisfying a certain recurrence relation. There are several works computing the discriminants of given polynomials. For example, Kaneko–Niiho and Mahlburg–Ono independently proved the formula for the discriminants of certain hypergeometric polynomials that are related to j-invariants of supersingular elliptic curves. Sawa–Uchida proved the formula for the discriminants of quasi-Jacobi polynomials and applied it to prove the nonexistence of certain rational quadrature formulas. Our main theorem presents a uniform way to prove a vast generalization of the above formulas for the discriminants. [ABSTRACT FROM AUTHOR]

In this paper, we study that the algebraic multiplicity of the zero Laplacian eigenvalue of a connected uniform hypergraph. We give the algebraic multiplicity of the zero Laplacian eigenvalue of a hyperstar. For a loose hyperpath, we characterize the algebraic multiplicity of the zero Laplacian eigenvalue by the multiplicities of points in the affine variety defined by the Laplacian eigenvalue equations. We compute the algebraic multiplicities of the zero Laplacian eigenvalue of a loose hyperpath. We also show that the algebraic multiplicity of the zero Laplacian eigenvalue is not smaller than the number of irreducible components of the eigenvariety associated with the zero Laplacian eigenvalue for a loose hyperpath. [ABSTRACT FROM AUTHOR]

Inverse Dynamics is routinely used in biomechanics for the estimation of loading in the musculoskeletal system but there are problems with the terms and definitions and even official recommendations introduce artificial and incorrect mechanical constructs to justify arbitrary and inappropriate terms. These terminology problems lead to further confusion and misinterpretations rather than to standardisation of mechanically correct nomenclature and accurate interpretation of joint loading. The perspective in this paper exposes some of the flawed foundational premises of these constructs and makes recommendations for accurate reporting of inverse dynamics outcomes and musculoskeletal loading. The inverse dynamics approach is based on free body diagrams that include the actual forces as applied ('Actual Forces' approach) or the replacement of actual forces with an equivalent resultant force and moment ('Resultant Moments' approach). Irrespective of the approach used to model the muscle and other forces, the inverse dynamics outputs always include the joint reaction forces representing the interactions with adjacent segments. The different terms suggested to distinguish the calculated joint reaction forces from the two approaches such as 'net joint force', 'resultant force', 'intersegmental force' and 'bone-on-bone force' are inappropriate, misleading and confusing. It is recommended to refer to joint reaction forces as Total or Partial when using an Actual Forces or a Resultant Moments approach, respectively. [ABSTRACT FROM AUTHOR]

An approach to calculating sums of some types of multiple numerical series is presented. This approach is based on using the formula for the resultant of a polynomial (or an entire function with a finite number of zeros) and an entire function obtained earlier by A.M. Kytmanov and E.K. Myshkina. This formula does not require values of the roots of the functions under study and is a combinatorial expression. By calculating the resultant of a polynomial and an entire function in two different ways, it is possible to obtain a relation for multiple numerical series. For the second way to find the resultant, we use the product of one function at the roots of another. In this article, the sums of some types of multiple numerical series that were previously absent in known reference books are found. They are expressed in terms of well-known special functions such as the Bessel function. This approach to calculating the sums of multiple numerical series differs significantly from the method based on the use of residue integrals associated with a system of equations. The relevance of this problem is determined by the fact that in applied problems, for example, in the equations of chemical kinetics, there are functions and systems of equations consisting of exponential polynomials.

The article is devoted to investigation of simple and multiple roots of systems of transcendental equations. It is shown that the number is related to the number of real roots of the resultant of the system. Examples for systems of equations are given.

In this work, we establish formulas for the prime decompositions of resultant and discriminant of the ring class polynomial associated with Thompson series for Fricke group Γ 0 (p) + for p prime and imaginary quadratic field, and as a consequence, validate a conjecture of Chen and Yui on the upper bound of prime factors of such resultants and discriminants. All associated numerical conjectures of Chen and Yui are verified using the Magma code for our formulas written by Chao Qin. [ABSTRACT FROM AUTHOR]

Computer algebra methods are used to investigate the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves along a circular orbit under the action of gravitational torque in the plane of the orbit. An algebraic method based on the resultant approach is applied to reduce the satellite stationary motion system of algebraic equations to a single algebraic equation in one variable, which determines the equilibrium configurations of the two-body system in the orbital plane. Classification of domains with equal numbers of equilibrium solutions is carried out using algebraic methods for constructing discriminant hypersurfaces. Discriminant curves in the space of system parameters that determine boundaries of domains with a fixed number of equilibria for the two-body system are obtained symbolically. Using the proposed approach it is shown that the satellite-stabilizer system can have up to 24 equilibrium orientations in the plane of a circular orbit. Some simple cases of the problem were studied in detail. [ABSTRACT FROM AUTHOR]

Late 18th and early 19th century Japanese mathematicians (wasanka) found solutions of two problems concerning the incircles of the quarter-triangles and skewed sectors of cyclic quadrilaterals. There is a modern proof of the first solution, but it makes extensive use of trigonometry and is therefore unlikely to be what a wasanka would have written. As for the second solution, Aida Yasuaki (1747–1817) gave two proofs for it, the second of which has been summarized in Japanese, but not the first. All three proofs are presented here together with commentary on their mathematical and historical significance. [ABSTRACT FROM AUTHOR]

This study aims to enhance precise analytical methods for image construction from circular sources in gravitational lenses. We begin by formulating the lens equations, using the bending angle formula for a light ray in a single-point lens with weak deflection and the thin lens approximation. We delve into the mathematical concepts of the resultant and Sylvester matrix, which help eliminate variables and find common polynomial roots. These concepts aid in parameter elimination within the lens equations to determine image coordinates. We showcase our method's application in one-point and binary lenses. In both cases, we derive implicit analytical expressions that entirely define the images of a circular source at any given position. By advancing analytical techniques, we reduce the reliance on numerical simulations, which can be computationally intensive and may not guarantee the discovery of all solutions. This work contributes to the understanding and characterization of gravitational lenses and provides a foundation for further developments in the field. [ABSTRACT FROM AUTHOR]

In nice cases, a zero-dimensional complete intersection ideal over a field has a Shape Lemma. There are also cases where the ideal is generated by the resultant and first subresultant polynomials of the generators. This paper explores the relation between these representations and studies when the resultant generates the elimination ideal. We also prove a Poisson formula for resultants arising from the hidden variable method. [ABSTRACT FROM AUTHOR]

We present two formulas for Chern classes (polynomial) of the tensor product of two vector bundles. In the first formula the Chern polynomial of the product is expressed as determinant of a polynomial in a matrix variable involving the Chern classes of the first bundle with Chern classes of the second bundle as coefficients. In the second formula the total Chern class of the tensor product is expressed as resultant of two explicit polynomials. Finally, formulas for the total Chern class of the second symmetric and the second alternating products are deduced.

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial s s p (N ∗) (X) for the Fricke group Γ 0 ∗ (N) , for N ∈ { 2 , 3 , 5 , 7 }. One of these conjectures gives a formula for the square of s s p (N ∗) (X) (mod p) in terms of a certain resultant, and the other relates the primes p for which s s p (N ∗) (X) splits into linear factors (mod p) to the orders of certain sporadic simple groups. [ABSTRACT FROM AUTHOR]

Objectives: To solve the problems in pedestrian dead reckoning algorithms for indoor positioning, the step recognition accuracy for step detection is not high, the synchronous control is not precise, and there is a large location deviation. Methods: An algorithm of improved finite state machine step detection for the activity of flat holding smartphone was proposed. A finite number of states corresponded to the trend of resultant acceleration variation during walking. Step detection and step cycle estimation were realized based on adjacent resultant acceleration difference and several thresholds of climbing and descending times. Two testers conducted experimental tests in 211 m corridors with flat holding smartphone, respectively. Results: Experimental results show that the accuracy of step detection tests is 100% by using the improved algorithm. It is 0.004 s earlier on average than the actual time for each step. Moreover, the average location error is 0.384 m. Compared to the auto‑correction analysis and acceleration differential based on finite state machine algorithms, the accuracy of step recognition, synchronous control, and location estimation are im ‑ proved by at least 0.7%, 60%, and 21.15%, respectively. Conclusions: The improved algorithm behaves better than the existing algorithms in step recognition, synchronous control, and location estimation. [ABSTRACT FROM AUTHOR]

We extend Quine's bound on the number of self-intersection of curves with polynomial parameterization to the case of Laurent polynomials. As an application, we show that circle embeddings are dense among all maps from a circle to a plane with respect to an integral norm. [ABSTRACT FROM AUTHOR]

Computer algebra methods are used to determine the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves in a central Newtonian force field on a circular orbit under the action of gravitational torque. Primary attention is given to the study of equilibrium orientations of the two-body system in the plane of the circular orbit. By applying symbolic differentiation, differential equations of motion are derived in the form of Lagrange equations of the second kind. A method is proposed for transforming the system of trigonometric equations determining the equilibria into a system of algebraic equations, which in turn are reduced by calculating the resultant to a single algebraic equation of degree 12 in one unknown. The roots of the resulting algebraic equation determine the equilibrium orientations of the two-body system in the circular orbit plane. By applying symbolic factorization, the algebraic equation is decomposed into three polynomial factors, each specifying a certain class of equilibrium configurations. The domains with an identical number of equilibrium positions are classified using algebraic methods for constructing a discriminant hypersurface. The equations for the discriminant hypersurface determining the boundaries of domains with an identical number of equilibrium positions in the parameter space of the problem are obtained via symbolic computations of the determinant of the resultant matrix. By numerical analysis of the real roots of the resulting algebraic equations, the number of equilibrium positions of the two-body system is determined depending on the parameters. [ABSTRACT FROM AUTHOR]

The paper presents certain aspects regarding the determination of the support of a sliding vector system resultant function of the resultant moment of the vector system. [ABSTRACT FROM AUTHOR]

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in K [ x ] . The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f (α) = f (β) for α , β in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras. [ABSTRACT FROM AUTHOR]

An explicit method to compute algebraic equations of curves of constant width and Zindler curves generated by a family of middle hedgehogs is given thanks to a property of Chebyshev polynomials. This extends the methodology used by Rabinowitz and Martinez-Maure in particular constant width curves to generate a full family of algebraic equations, both of curves of constant width and Zindler curves, defined by trigonometric polynomials as support functions. [ABSTRACT FROM AUTHOR]

We consider the identifiability problem for the parameters of series-parallel LCR circuit networks. We prove that for networks with only two classes of components (inductor-capacitor (LC), inductor-resistor (LR), and capacitor-resistor (RC)), the parameters are identifiable if and only if the number of non-monic coefficients of the constitutive equations equals the number of parameters. The notion of the "type" of the constitutive equations plays a key role in the identifiability of LC, LR, and RC networks. We also investigate the general series-parallel LCR circuits (with all three classes of components), and classify the types of constitutive equations that can arise, showing that there are 22 different types. However, we produce an example that shows that the basic notion of type that works to classify identifiability of two class networks is not sufficient to classify the identifiability of general series-parallel LCR circuits. [ABSTRACT FROM AUTHOR]

We use results concerning the Smith forms of circulant matrices to identify when cyclically presented groups have free abelianisation and so can be Labelled Oriented Graph (LOG) groups. We generalize a theorem of Odoni and Cremona to show that for a fixed defining word, whose corresponding representer polynomial has an irreducible factor that is not cyclotomic and not equal to ± t , there are at most finitely many n for which the corresponding n -generator cyclically presented group has free abelianisation. We classify when Campbell and Robertson's generalized Fibonacci groups H (r , n , s) are LOG groups and when the Sieradski groups are LOG groups. We prove that amongst Johnson and Mawdesley's groups of Fibonacci type, the only ones that can be LOG groups are Gilbert-Howie groups H (n , m). We conjecture that if a Gilbert-Howie group is a LOG group, then it is a Sieradski group, and prove this in certain cases (in particular, for fixed m , the conjecture can only be false for finitely many n). We obtain necessary conditions for a cyclically presented group to be a connected LOG group in terms of the representer polynomial and apply them to the Prishchepov groups. [ABSTRACT FROM AUTHOR]

Computer algebra methods were used to find the stationary motions of the system of two bodies connected by a spherical hinge, that moves along a circular orbit under the action of gravitational field. The resultant calculation approach was applied to reduce the stationary motion system of algebraic equations to a single algebraic equation in one variable that determines spatial equilibrium configurations of the two-body system. Classification of domains with equal numbers of equilibrium solutions was done using algebraic methods for constructing discriminant hypersurfaces. Bifurcation curves in the space of system parameters that determine boundaries of domains with a fixed number of equilibrium orientations of the two-body system were obtained symbolically. The number of equilibria was found by analyzing the real roots of the algebraic equations in the space of parameters of the problem. [ABSTRACT FROM AUTHOR]

When using resultants for elimination, one standard issue is that the resultant vanishes if the variety contains components of dimension larger than the expected dimension. J. Canny proposed an elegant construction, generalized characteristic polynomial, to address this issue by symbolically perturbing the system before the resultant computation. Such perturbed resultant would typically involve artefact components only loosely related to the geometry of the variety of interest. For removing these components, J.M. Rojas proposed to take the greatest common divisor of the results of two different perturbations. In this paper, we investigate this construction, and show that the extra components persistent under taking different perturbations must come either from singularities or from positive-dimensional fibers. [ABSTRACT FROM AUTHOR]

Given two polynomials a and b in F q [ x , y ] which have no non-trivial common divisors, we prove that a generator of the elimination ideal 〈 a , b 〉 ∩ F q [ x ] can be computed in quasi-linear time. To achieve this, we propose a randomized algorithm of the Monte Carlo type which requires (d e log ⁡ q) 1 + o (1) bit operations, where d and e bound the input degrees in x and in y respectively. The same complexity estimate applies to the computation of the largest degree invariant factor of the Sylvester matrix associated with a and b (with respect to either x or y), and of the resultant of a and b if they are sufficiently generic, in particular such that the Sylvester matrix has a unique non-trivial invariant factor. Our approach is to exploit reductions to problems of minimal polynomials in quotient algebras of the form F q [ x , y ] / 〈 a , b 〉. By proposing a new method based on structured polynomial matrix division for computing with the elements of the quotient, we succeed in improving the best-known complexity bounds. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the T-function, T(s), which is a polynomial defined by means of a univariate resultant constructed from a given parametrization of an algebraic space curve. It is shown that , where are polynomials (the fibre functions) whose roots are the fibre of the ordinary singularities of multiplicity. Therefore, a complete classification of the singularities of , via the factorization of a resultant, is obtained. [ABSTRACT FROM AUTHOR]

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Using Newton’s recurrent formulae, we find the product of values of an entire function of one variable in zeroes of another entire function. This allows to answer whether they have common zeros. By that, we propose an approach to construction of the resultant of two entire functions. We also give examples illustrating the main result.