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We investigate the number of straight lines contained in a K3 quartic surface \(X\) defined over an algebraically closed field of characteristic 3. We prove that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the Fermat quartic surface; otherwise, \(X\) contains at most 67 lines. We improve this bound to 58 if \(X\) contains a star (ie four distinct lines intersecting at a smooth point of \(X\)). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided., Comment: One example removed. Some examples and some proofs now with more details

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines., Comment: Final version accepted for publication. Theorem 1.1 had to be stated in a weaker version

In this paper, we prove that, over an algebraically closed field of odd characteristic p ( p ≠ 5 ), every supersingular K3 surface is isomorphic to a quartic surface in the three dimensional projective space .

We describe a set of generators and defining relations for the group of birational automorphisms of a general 15-nodal quartic surface in the complex projective 3-dimensional space.

We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines and special lines. We answer a question by Oguiso on a determinantal presentation of the Fermat quartic surface., 22 pages, the last unknown quartic surface with more than 52 lines has been added

Mustaţa has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P 3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.

We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics., Comment: The paper reflects the considerable development (explicit equations by X. Roulleau and B. Naskrecki) that followed the original version

We study the surface of Gauss double points associated to a very general quartic surface and the natural morphisms associated to it., 20 pages, 7 figures

With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $\gamma$ generates $\Fqn$ over $\Fq$ and $\beta$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that $\beta(\gamma + a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$., Comment: To appear in Math. Comp

We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in Kolluru et al., (Appl. Algebra Engrg. Comm. Comput. 10(6):433–464, 2000, Ex. 3.2). Among the codes that we construct almost all have parameters as good as the best known codes according to Grassl (2007) and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound (Geil and Høholdt, IEEE Trans. Inform. Theory 46(2), 635–641, 2000 and Høholdt 1998) from Gröbner basis theory and for this purpose we develop a new method where we inspired by Buchberger’s algorithm perform a series of symbolic computations.

We construct three elements in the kernel of the tame symbol on families of quartic curves. We show that these elements are integral under certain conditions on the parameters. Moreover, we prove that these elements are in general linearly independent by calculating the limit of the regulator.

We derive the quartic interaction vertex of pure Yang-Mills theory by demanding closure of the light-cone Poincar\'e algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We then prove that corrections to the spin generator, for spin one at this order, vanish. We comment briefly on higher spin fields in this context., Comment: 10 pages. v2: minor changes, references added

We study smooth quartic surfaces in ℙ3which admit a group of projective automorphisms isomorphic to the icosahedron group.

The aim of this paper is first to reformulate the fixed point theorem [4, Theorem 1] in 2-Banach spaces, after it, we introduce and solve the radical quartic functional equation f ( x 4 + y 4 4 ) = f ( x ) + f ( y ) . We also show that the fixed point methods allow to investigate Ulam's type stability of radical quartic functional equation in 2-Banach spaces.

The previous approximation curves of conic section by quartic Bzier curves interpolate the conic section at the specified parameter values. In this paper, by solving the minimax problem, we present the optimal parameter values for approximating conic sections by the quartic Bzier curves. The upper bound on the Hausdorff distance between the conic section and the approximation curves is minimized. The method of solving the minimax problem is changed to solve a quartic algebraic equation by delicate reasoning.

Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Gruji\'c and Kalisch., Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1706.04659

Let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G, In this new version we fix some errors in Table 5

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{{\rm sa}}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such that the boundary $\partial W$ is a strictly normal crossing divisor. The main part of the L-function is described by some elliptic newform $g$ of weight 3. Then we construct an automorphic representation $\Pi$ of ${\rm GSp}_2(\A)$ related to $g$ and an explicit vector $E_Z$ sits inside $\Pi$ which creates a vector valued (non-cuspidal) Siegel modular form of weight $(3,1)$ so that $F_Z$ coincides with $E_Z$ in $H^{2,0}(\partial W)$ under the Poincar\'e residue map and various identifications of cohomologies.

The paper deals with the construction of bivariate quartic splines on Powell–Sabin triangulations. In particular, it provides a spline space that is C 2 everywhere except across some edges of the refined triangulation. The splines that belong to this space can be described uniquely with interpolation values and derivatives of order at most two. Moreover, they can be represented in a locally supported basis that forms a convex partition of unity. As an application of this result, quasi-interpolants that reproduce polynomials of degree four are derived.

We show that there cannot be more than 64 lines on a quartic surface admitting isolated rational double points over an algebraically closed field of characteristic $p \neq 2,\,3$, thus extending Segre--Rams--Sch\"utt theorem. Our proof offers a deeper insight into the triangle-free case and takes advantage of a special configuration of lines, thereby avoiding the technique of the flecnodal divisor. We provide several examples of non-smooth K3 quartic surfaces with many lines., Comment: 31 pages, 3 tables, 1 figure

We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory., Comment: v2: Significant changes at the request of the referee. Section 3 has been rearranged to accommodate a revised proof of Proposition 3.5 (formerly 3.2). Section 5 has been removed completely, it will instead appear as part of Section 5 in arxiv:1601.08110

In this paper, we present a C 3 quartic B-spline approximation of circular arcs. The Hausdorff distance between the C 3 quartic B-spline curve and the circular arc is obtained in closed form. Using this error analysis, we show that the approximation order of our approximation method is six. For a given circular arc and error tolerance we find the C 3 quartic B-spline curve having the minimum number of control points within the tolerance. The algorithm yielding the C 3 quartic B-spline approximation of a circular arc is also presented.

We study exact multiplicity and bifurcation curves of positive solutions of the boundary value problem \begin{eqnarray} &u"(x)+\lambda (-u^4+\sigma u^3-\tau u^2+\rho u)=0, -1 < x < 1, \\ &u(-1)=u(1)=0, \end{eqnarray} where $\sigma, \tau \in \mathbb{R}$, $\rho \geq 0,$ and $\lambda >0$ is a bifurcation parameter. Then on the $(\lambda, \|u\|_\infty)$-plane, we give a classification of four qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, a $\subset$-shaped curve and a monotone increasing curve.

We provide explicit evaluations of the Chowla-Selberg formula for quartic abelian CM fields due to Barquero-Sanchez and Masri. These identities relate values of a Hilbert modular function at CM points to values of Euler’s gamma function Γ \Gamma and an analogous function Γ 2 \Gamma _2 at rational numbers. Our work consists of two main parts. First, we implement an algorithm in SageMath to compute the CM points. Second, we exhibit families of quartic abelian CM fields for which the part of the formula involving values of Γ \Gamma and Γ 2 \Gamma _2 takes a particularly simple form.

There are seven time independent, integrable, Henon-Heiles systems: three with cubic and four with quartic potential. The cubic and one of the quartic cases have been separated in the last decades. The other three cases 1:6:1, 1:6:8 and 1:12:16 have resisted several attempts in the last years. In this paper we focus our attention on the 1:12:16 case whose equations of motion have been separated only in the degenerate case ab = 0. We give here the separation coordinates for the generic case using a method introduced by Franco Magri in 2005 under the name of Kowalevski’s Conditions.

The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the N\'eron-Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point free involution that gives rise to the Enriques surface of type IV in Nikulin-Kondo-Martin's classification of Enriques surfaces with finite automorphism group. We investigate the specialization of this involution to characteristic 3., Comment: 32 pages. The proofs in Section 6 are simplified. The main results are not changed

In [1] we proved the stability of quartic functional equations by using 2-variable quadratic functional equations inLipschitz spaces. In this paper,we show that the results presented in [1] only hold for quadratic functional equations and sowe correct our results and approximate the quartic functional equations by using biquadratic functional equations. Indeed, the equality K (2x, 2y) = 24K (x, y), used in [1, Theorem 2.1], does not hold for the 2-variable quadratic functional equation, that is, the function Q defined in [1, Theorem 2.1] is quadratic and it is not quartic. We need to introduce the notion of symmetric left invariant mean. Let G be an abelian group, E a vector space, and S(E) a family of subsets of E . By B(G, S(E))we denote the family of all functions f : G −→ E such that Im f ⊂ A for some A ∈ S(E). We say that B(G, S(E)) admits a symmetric left invariant mean (briefly SLIM), if the family S(E) is linearly invariant and there exists a linear operatorM : B(G, S(E)) −→ E such that (i) if fx,y ∈ B(G, S(E)) and (x, y) ∈ G × G, then M[ fx,y] = M[ fy,x ]

In this paper, the relations between p-ranks of the tame kernel and the ideal class group for a general number field are investigated. As a result, nearly all of Browkin's results about quadratic fields are generalized to those for general number fields. In particular, a p-rank formula between the tame kernel and the ideal class group for a totally real number field of odd degree is obtained when p is a Fermat prime. As an example, the case of cyclic quartic fields is considered in more details. More precisely, using the results on cyclic quartic fields, we give some results connecting the p-rank(K2OF) with the p-rank(Cl(OE)), where F is a cyclic quartic field and E is an appropriate subfield of F(ζp).

Historically, much of the theory and practice in nonlinear optimization has revolved around the quadratic models. Though quadratic functions are nonlinear polynomials, they are well structured and easy to deal with. Limitations of the quadratics, however, become increasingly binding as higher degree nonlinearity is imperative in modern applications of optimization. In the recent years, one observes a surge of research activities in polynomial optimization, and modeling with quartic or higher order polynomial functions has been more commonly accepted. On the theoretical side, there are also major recent progresses on polynomial functions and optimization. For instance, Ahmadi et al. [2] proved that checking the convexity of a quartic polynomial function is strongly NP-hard in general, which settles a long-standing open question. In this paper we proceed to studying six fundamentally important convex cones of quartic functions in the space of symmetric quartic tensors, including the cone of nonnegative quartic polynomials, the sum of squared polynomials, the convex quartic polynomials, and the sum of fourth powered polynomials. It turns out that these convex cones coagulate into a chain in decreasing order. The complexity status of these cones is sorted out as well. Finally, potential applications of the new results to solve highly nonlinear and/or combinatorial optimization problems are discussed.

We present an explicit representation of a general C 2 rational quartic interpolation spline with three families of local shape parameters. Sufficient conditions for the proposed interpolant to preserve locally the shape of positive, monotonic, and/or convex set of data are given. The approximation order of the interpolant is O ( h 2 ) or O ( h 3 ) .

Over a field k of characteristic 3, we prove that there are no geometrically smooth quartic surfaces SP 3 with more than 112 lines. Moreover, the surface with 112 lines is projectively equivalent overto the Fermat quartic. As a key ingredient, we derive a characteristic free upper bound for the number of lines met by a quadric on a smooth quartic surface.

We construct infinite families of elliptic curves with cyclic torsion groups over quartic number fields K such that the Galois closure of K is dihedral of degree 8; such a quartic number field K is called a dihedral quartic number field. In fact, all the cyclic torsion groups of elliptic curves which occur over quartic number fields (but not over quadratic number fields) are Z/NZ with N=17,20,21,22,24. The cases of N=20 and 24 are treated in the previous work of the authors, and the current work completes the construction of infinite families of elliptic curves over dihedral quartic number fields with cyclic torsion groups (which do not occur over quadratic number fields).

Let [math] be the family of complex-valued polynomials of the form [math] with [math] . The Gauss–Lucas theorem guarantees that the critical points of [math] will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, the unit disk contains two “desert” regions, the open disk [math] and the interior of [math] , in which critical points of [math] cannot occur. Furthermore, each [math] inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in [math] .

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras., 29 pages

This paper proposes a new approximation method for conic sections by quartic Bezier curves with G 1 end-point continuity. We give an upper bound on the Hausdorff distance between the conic section and the quartic Bezier approximation curve, and also show that the approximation order is eight. Moreover, using the subdivision scheme at the shoulder point of the conic section, the resulting approximation curve has G 2 (curvature)-continuity with the conic section at the endpoints. Finally, we prove that our approximation has a smaller error bound than previous quartic Bezier approximations, and also present some numerical examples to demonstrate the validity and effectiveness of our method.

Given a quartic field with S 4 Galois group, we relate its ramification to that of the non-Galois sextic subfields of its Galois closure, and we construct explicit generators of these sextic fields from that of the quartic field, and vice versa. This allows us to recover examples of S 4 -sextic fields of Cohen and of Tate unramified outside 229, and to easily determine the tame part of the conductor of an octahedral Artin representation. We study class number divisibility arising from S 4 -quartics whose discriminants are odd and square-free, we explicitly construct infinitely many S 4 -quartics whose discriminants are −1 times a square, and experimental data suggest two surprising conjectures about S 4 -quartic fields over Q unramified outside one finite prime.

In this paper, we will evaluate the values of the Dedekind zeta functions at s = − 1 of the simplest quartic fields. We first introduce Siegel's formula for the values of the Dedekind zeta function of a totally real number field at negative odd integers, and will apply Siegel's formula to the simplest quartic fields. In the second, we will develop the basic arithmetic properties of the simplest quartic fields which will be necessary in our computation. We will compute the discriminant, ring of integers, and different of the simplest quartic fields. In the third, we will give a full description for a Siegel lattice of the simplest quartic fields, and develop a method of computing sum of divisors function for an ideal. Finally, by combining these results, we compute the values of the Dedekind zeta functions at s = − 1 of the simplest quartic fields.

In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition $abcd\neq \square$. In particular, we present an infinite family of diagonal quartic surfaces defined over $\Q$ with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type $ax^6+by^6+cz^6+dw^i=0$, $i=2$, $3$, or $6$, with infinitely many rational points., Comment: revised version will appear in International Journal of Number Theory

A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula., Minor typos corrected. To appear on Proceedings of the American Mathematical Society

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions in n variables are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009, and the first substantial progress on the much more complicated cubic case came in 2010. In this paper, we show that much of the work on the cubic case can be extended to the quartic case. We also prove an exact formula for the number and sizes of the affine equivalence classes when n is a prime.

The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space CP^3., Comment: Some explanation added in section 1, other minor amendments and reformatting; to appear in JEMS

Related to the components of the quaternionic Hopf mapping, we propose a parametric Hamiltonian function in $\mathbb{T}^*\mathbb{R}^4$ which is a homogeneous quartic polynomial with six parameters, defining an integrable family of Hamiltonian systems. The key feature of the model is its nested Hamiltonian-Poisson structure, which appears as two extended Euler systems in the reduced equations. This is fully exploited in the process of integration, where we find two 1-DOF subsystems and a quadrature involving both of them. The solution is quasi-periodic, expressed by means of Jacobi elliptic functions and integrals, based on two periods. For a suitable choice of the parameters, some remarkable classical models such as the Kepler, geodesic flow, isotropic oscillator and free rigid body systems appear as particular cases.

In this paper, we obtain the general solution and investigate its generalized Hyers-Ulam-Rassias stability of a new generalized cubic and quartic functional equation ()

In this paper we present a method of circular arc approximation by quartic Bezier curve. Our quartic approximation method has a smaller error than previous quartic approximation methods due to the alternation of the error function of our quartic approximation. Our method yields a closed form of error so that subdivision algorithm is available, and curvaturecontinuous quartic spline under the subdivision of circular arc with equal-length until error is less than tolerance. We illustrate our method by some numerical examples.

We test the methods for computing the Picard group of a K3 surface in a situation of high rank. The examples chosen are resolutions of quartics in P 3 having 14 singularities of type A 1. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.