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In this paper, we introduce new plane curves, i.e. Cassinian focal-directorial curves as curves of a transitory type, of Cassinian multidirectorial curves as counterpart to Cassinian multifocal curves; and demonstrate their geometric genesis, and their form variation for special choices of foci and directrices. The form variation of Cassinian focal-directorial curves (CFDC) for different values of the particular parameter is also considered. Furthermore, it is shown that these algebraic curves also might have applications in architectural space design. [ABSTRACT FROM AUTHOR]

We revisit the seminal Brill–Noether algorithm for plane curves with ordinary singularities. Our new approach takes advantage of fast algorithms for polynomials and structured matrices. We design a new probabilistic algorithm of Las Vegas type that computes a Riemann–Roch space in expected sub-quadratic time. [ABSTRACT FROM AUTHOR]

In this paper, we study the algebraic limit cycle of degree 2 for the system x ˙ = - y + a 30 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 , y ˙ = x + b 30 x 3 + b 21 x 2 y + b 12 x y 2 + b 03 y 3 , which is called a linear type center plus cubic homogeneous polynomial differential system. We derive the normal form of this system with an algebraic limit cycle of degree 2. It is shown that, if an invariant algebraic curve of degree 2 is a limit cycle of the above mentioned system, then it is a unique limit cycle in the phase plane. [ABSTRACT FROM AUTHOR]

We study the topology of the moduli space of unramified Z / p -covers of tropical curves of genus g ≥ 2 , where p is a prime number. We use recent techniques by Chan–Galatius–Payne to identify contractible subcomplexes of the moduli space. We then use this contractibility result to show that this moduli space is simply connected. In the case of genus 2, we determine the homotopy type of this moduli space for all primes p. This work is motivated by prospective applications to the top-weight cohomology of the space of prime cyclic étale covers of smooth algebraic curves. [ABSTRACT FROM AUTHOR]

Elliptic curves are considered highly secure due to their complex mathematical structure and are widely used in cryptographic key exchange algorithms. This study aimed to construct a confusion component for a symmetric block cipher using a hybrid approach that involves elliptic curve points and an algebraic group structure. Generally, elliptic curve points do not generate much randomness, so they are not recommended for constructing a highly nonlinear S-box. In this study, we first randomized the elliptic curve points to the maximum possible level and generate initial S-boxes. These initial S-boxes were then evolved through algebraic group action to enhance their nonlinearity. The strength of the proposed substitution box was assessed using various tests, including nonlinearity, bit independence criterion, strict avalanche criterion, and differential and linear probabilities. Using this approach, we designed an S-box with a nonlinearity value of 114, which is higher than the standard value presented by AES. We compared the results of other cryptographic tests with well-known S-boxes to validate the effectiveness of the suggested confusion component. [ABSTRACT FROM AUTHOR]

Several applications of function fields over finite fields, or equivalently, algebraic curves over finite fields, require computing bases for Riemann–Roch spaces. In this paper, we determine explicit bases for Riemann–Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized algebraic geometry codes with good parameters. [ABSTRACT FROM AUTHOR]

Let G be a reductive group, and let X be an algebraic curve over an algebraically closed field k with positive characteristic. We prove a version of nonabelian Hodge correspondence for tame G-local systems over X and logarithmic G-Higgs bundles over the Frobenius twist X ′ . To obtain a full description of the correspondence for the noncompact case, we introduce the language of parahoric group schemes to establish the correspondence. [ABSTRACT FROM AUTHOR]

For a , b , c , d , e ∈ Q × , denote by C a , b , c , d , e the nonsingular projective curve over Q given by the affine equation C a , b , c , d , e : a x 2 + b y 2 = c + d x y + e x 2 y 2. In this paper, we find a relation between the number of points on C a , b , c , d , e over F q and the finite field Appell series F 4 ∗ . As a consequence, in view of certain transformations for finite field Appell series F 4 ∗ , we prove the existence of isogenies between C a , b , c , d , ab c and the twisted Edward family of elliptic curves E α , 16 a b α d 2 . [ABSTRACT FROM AUTHOR]

A number field K is primitive if K and Q are the only subextensions of K. Let C be a curve defined over Q . We call an algebraic point P ∈ C (Q ¯) primitive if the number field Q (P) is primitive. We present several sets of sufficient conditions for a curve C to have finitely many primitive points of a given degree d. For example, let C / Q be a hyperelliptic curve of genus g, and let 3 ≤ d ≤ g - 1 . Suppose that the Jacobian J of C is simple. We show that C has only finitely many primitive degree d points, and in particular it has only finitely many degree d points with Galois group S d or A d . However, for any even d ≥ 4 , a hyperelliptic curve C / Q has infinitely many imprimitive degree d points whose Galois group is a subgroup of S 2 ≀ S d / 2 . [ABSTRACT FROM AUTHOR]

We study moduli stacks of principal C ∗ -bundles over nodal complex algebraic curves and determine their rational cohomology algebras in terms of Chern classes. [ABSTRACT FROM AUTHOR]

We show that on a metric graph of genus g, a divisor of degree n generically has g (n - g + 1) Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass points become distributed according to the Zhang canonical measure. In other words, the limiting distribution is determined by effective resistances on the metric graph. This distribution result has an analogue for complex algebraic curves, due to Neeman, and for curves over non-Archimedean fields, due to Amini. [ABSTRACT FROM AUTHOR]

(N , γ) -hyperelliptic semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated with totally ramified points of N-fold covers of curves of genus γ . Torres characterized (2 , γ) -hyperelliptic semigroups of maximal weight whenever their genus is large relative to γ . Here we do the same for (3 , γ) -hyperelliptic semigroups, and we formulate a conjecture about the general case whenever N ≥ 3 is prime. [ABSTRACT FROM AUTHOR]

The study of formal stability of equilibrium positions of a multiparametric Hamiltonian system in a generic case is traditionally carried out using its normal form under the condition of the absence of resonances of small orders. In this paper we propose a method of symbolic computation of the condition of existence of a resonance of arbitrary order for a system with three degrees of freedom. It is shown that this condition for each resonant vector can be represented as a rational algebraic curve. By methods of computer algebra the rational parametrization of this curve for the case of an arbitrary resonance is obtained. A model example of some two-parameter system of pendulum type is considered. [ABSTRACT FROM AUTHOR]

In arithmetic and algebraic geometry, superspecial (s.sp. for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If g ≥ 4 , it is not even known whether there exists such a curve of genus g in general characteristic p > 0 , and in the case of g = 4 , several computational approaches to search for those curves have been proposed. In the genus-4 hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp. curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group as a subgroup. In this paper, we propose an algorithm with complexity O ~ (p 4) in theory but O ~ (p 3) in practice to enumerate s.sp. hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order 6. By executing the algorithm over Magma, we enumerate those curves for p up to 1000. We also succeeded in finding a s.sp. hyperelliptic curve of genus 4 in every p with p ≡ 2 (mod 3) . [ABSTRACT FROM AUTHOR]

For a long time, it was believed that Alfred Kempe proved (1876) a theorem about the possibility of plotting an arbitrary planar algebraic curve by parts using hinge mechanisms (linkages). However, at the end of the 20th century, specialists in algebraic geometry, having rediscovered and developed this result in modern language, began to assert that Kempe's reasoning contained significant gaps and errors. In the author's opinion, these claims are unfounded. In this paper, by a detailed analysis of the notions of a linkage and its configuration space, the author tries to substantiate this point of view. [ABSTRACT FROM AUTHOR]

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over F p of degree at most 4 p 3 / 4 + 1 whose points form blocking sets. [ABSTRACT FROM AUTHOR]

Given a finite set of points S ⊂ R d , a k-set of S is a subset A ⊂ S of size k which can be strictly separated from S \ A by a hyperplane. Similarly, a k-facet of a point set S in general position is a subset Δ ⊂ S of size d such that the hyperplane spanned by Δ has k points from S on one side. For a probability distribution P on R d , we study E P (k , n) , the expected number of k-facets of a sample of n random points from P. When P is a distribution on R 2 such that the measure of every line is 0, we show that E P (k , n) = O (n (k + 1) 1 / 4) . Our argument is based on a technique by Bárány and Steiger. We study how it may be possible to improve this bound using the continuous version of the polynomial partitioning theorem. This motivates a question concerning the points of intersection of an algebraic curve and the k-edge graph of a set of points. We also study a variation on the k-set problem for the set system whose set of ranges consists of all translations of some strictly convex body in the plane. The motivation is to show that the technique by Bárány and Steiger is tight for a natural family of set systems. For any such set system, we determine bounds for the expected number of k-sets which are tight up to logarithmic factors. [ABSTRACT FROM AUTHOR]

We study random domino tilings of the Aztec diamond with a biased 2 × 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight. [ABSTRACT FROM AUTHOR]

In the paper Targino (Outer Lipschitz Geometry of Complex Algebraic Plane Curves. International Mathematics Research Notices, rnac202. https://doi.org/10.1093/imrn/rnac202, 2022) Renato Targino shows that bi-Lipschitz type of a plane curve is determined by the local ambient topological properties of curves. Here we show that it is no longer true in higher dimensions. However we show that the bi-Lipschitz type of space curves is determined by the number of singular points and by the local ambient topological type of a generic projection of such curves into the affine plane. [ABSTRACT FROM AUTHOR]

In this paper we present two algorithms for computing the g-asymptotes or generalized asymptotes, of a plane algebraic curve, C , implicitly or parametrically defined. The asymptotes of a curve C reflect the status of C at points with sufficiently large coordinates. It is well known that an asymptote of a curve C is a line such that the distance between C and the line approaches zero as they tend to infinity. However, a curve C may have more general curves than lines describing the status of C at infinity. These curves are known as g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented, as well as the corresponding implementations. For this purpose, we use the algebra software Maple. A comparative analysis of the algorithms is carried out, based on some properties of the input curves and their results to analyze the efficiency of the algorithms and to establish comparative criteria. The results presented in this paper are a starting point to generalize this study to surfaces or to curves defined by a non-rational parametrization, as well as to improve the efficiency of the algorithms. Additionally, the methods developed can provide a new and different approach in prediction (regression) or classification algorithms in the machine learning field. [ABSTRACT FROM AUTHOR]

The substitution box (S-box) is the only nonlinear component of any block cipher. It hides the relationship between the key and the cipher text. Several methods are available in the literature for creating S-boxes of different sizes. However, the need to design 4 × 4 S-boxes for lightweight encryption is in demand. The suggested method first generates the initial S-boxes from the elliptic curve points, and then the S-boxes are evolved with the help of algebraic group structures. Algebraic group structures have the potential to improve their cryptographic strength. The cryptographic properties of the suggested S-boxes, such as nonlinearity, algebraic degree, branch number, and bijectivity, are analyzed. We also present a comparative analysis of the suggested S-boxes with different standard S-boxes. The offered S-boxes are utilized in an image steganography scheme. First, the secret image is encrypted through the combination of the substitution operation and the bitwise XOR of the key. The enciphered image is hidden in the cover image using the least significant bit of steganographic scheme. Experimental results show that the suggested algorithm has a high security level and better stego-image quality. [ABSTRACT FROM AUTHOR]

Let F q 2 be the finite field with q 2 elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over F q 2 with many rational points. The curves constructed are Kummer covers or fiber products of Kummer covers of the projective line. Further, we compute the exact number of rational points for some of the curves. [ABSTRACT FROM AUTHOR]

In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve X were investigated and the sets of minimal generators were determined for all points in X (F q 2) and X (F q 6) \ X (F q 2) . This paper completes their work by settling the remaining cases, that is, for points in X (F ¯ q) \ X (F q 6) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in X (F q 7) \ X (F q) and we give a bound on the Feng–Rao minimum distance d ORD . For q = 3 we provide a table that also reports the exact values of d ORD . As a further application we construct quantum codes from F q 7 -rational points of the GK-curve. [ABSTRACT FROM AUTHOR]

Combinatorial t-designs have nice applications in coding theory, finite geometries and engineering areas. t-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs t-designs from the quadratic polynomial x 4 + x 3 over GF (2 n) and determine their parameters. We yield 2- 2 n , 3 · 2 n - 2 , 3 · 2 n - 2 3 · 2 n - 2 - 1 designs for n even and 3- 2 n , 2 n - 1 , 2 n - 1 2 n - 2 - 1 designs for n odd. [ABSTRACT FROM AUTHOR]

A method for symbolic computation of a condition of existence of a third- and fourth-order resonance for investigations of formal stability of an equilibrium state of a multiparameter Hamiltonian system with three degrees of freedom in the case of general position is proposed. This condition is formulated in the form of zeros of a quasi-homogeneous polynomial of the coefficients of the characteristic polynomial of the linear part of the Hamiltonian system. Computer algebra (Gröbner bases of elimination ideals) and power geometry (power transformations) are used to represent this condition for various resonance vectors in the form of rational algebraic curves. Given a linear approximation of the characteristic polynomial in the space of its coefficients, these curves are used to obtain a description of a partition of the stability domain into parts in which there are no strong resonances. An example of a description of resonance sets for a two-parameter pendulum-type system is given. All computations are carried out in the computer algebra system Maple. [ABSTRACT FROM AUTHOR]

The theory of commutative rings of differential operators relating the completely integrable systems with the geometry of algebraic curves was constructed several decades ago. It was especially complete in the case of rings, generated by two operators of the coprime order, usually of order 2 and of some odd order; the theory of such rings turned out to be equivalent to the theory of KdV hierarchy. However, the corresponding algebraic curves were always hyperelliptic. In order to handle the general (canonical curves), one should consider the rings, generated by more than two operators. In the previous paper of 1980, the author considered the simplest possible case of this kind—that of generators of orders 3, 4, 5. The goal of the present paper is to give the details of the calculations in that paper and to explain the conjectural geometry underlying some enigmatic phenomena that were used in 1980 to complete the calculations and give some algebro-geometric applications. [ABSTRACT FROM AUTHOR]

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension n admitting real-fibered morphisms to the n-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles. [ABSTRACT FROM AUTHOR]

Let G be a semiabelian variety and C a curve in G that is not contained in a proper algebraic subgroup of G. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the so-called unlikely intersections of C with subgroups of codimension at least 2. In this note, we establish this assertion for general semiabelian varieties over Q ¯ . This extends results of Maurin and Bombieri, Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in the abelian case. [ABSTRACT FROM AUTHOR]

Application of the method of computing the location of all types of level lines of a real polynomial on the real plane is demonstrated. The theory underlying this method is based on methods of local and global analysis by the means of power geometry and computer algebra. Three nontrivial examples of computing level lines of real polynomials on the real plane are discussed in detail. The following computer algebra algorithms are used: factorization of polynomials, computation of the Gröbner basis, construction of the Newton polygon, and representation of an algebraic curve on a plane. It is shown how computational difficulties can be overcome. [ABSTRACT FROM AUTHOR]

Precise tracking of a point-target on a nonlinear trajectory is challenging and has applications ranging from traffic analysis to microscopic particle tracking. To solve such a problem, we developed an algorithm which is independent of statistical-probabilistic and mechanical modeling, and free of analytical extrapolation methods. Our main objective was, to predict target's future location using its previous locations by a deep neural network, trained on a large data set of linear and nonlinear trajectories. To design our data-driven prediction approach, we developed a freely available database of up to second-order algebraic curves uniformly distributed in a given domain. This database could be used for training and testing point-target tracking algorithms. Simulated noisy test sets of trajectories were produced using Gaussian noise for analyzing the forecasting performance and noise sensitivity of our model. Further, the newly designed long short-term memory-based network that uses polar coordinates for its training is capable of predicting the target's future locations on real-world smooth trajectories. We compared the proposed predictor network to classical and state-of-the-art predictors based on average absolute and relative errors. The experimental results demonstrated that our novel predictor achieved up to 47% improvement on test data sets. The observed area under the noise response curve has improved by up to 11%. [ABSTRACT FROM AUTHOR]

Let K r be the complete elliptic integral of the first kind defined on 0 , 1 . By virtue of the auxiliary function H f , g = f ′ / g ′ g - f , we prove that the function Q p x = ln p / 1 - x K x is strictly convex on 0 , 1 if and only if 0 < p ≤ 4 , thus answering a conjecture. Moreover, we completely described the monotonicity of Q p x on 0 , 1 for different p ∈ 0 , ∞ . [ABSTRACT FROM AUTHOR]

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues k > θ 1 > θ 2 > θ 3 , and that the triple (θ 1 , θ 2 , θ 3) satisfies a certain homogeneous polynomial equation of degree s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for s = 5 and s = 7 , there are only the obvious solutions, and we conjecture this to remain true for all (odd) s ≥ 9 . [ABSTRACT FROM AUTHOR]

Gauss's class number one problem, solved by Heegner, Baker, and Stark, asked for all imaginary quadratic fields for which the ideal class group is trivial. An application of this solution gives all elliptic curves that can be defined over the rationals and have a large endomorphism ring (CM). Analogously, to get all CM curves of genus two defined over the smallest number fields, we need to find all quartic CM fields for which the CM class group (a quotient of the ideal class group) is trivial. We solve this CM class number one problem. We prove that the list given in Bouyer–Streng [LMS J Comput Math 18(1):507–538, 2015, Tables 1a, 1b, 2b, and 2c] of maximal CM curves of genus two defined over the reflex field is complete. We also prove that there are exactly 21 simple CM curves of genus two over C that can be defined over Q . [ABSTRACT FROM AUTHOR]

A closed Riemann surface S is called a generalized Fermat curve of type (p, n), where n , p ≥ 2 are integers such that (p - 1) (n - 1) > 2 , if it admits a group H ≅ Z p n of conformal automorphisms with quotient orbifold S/H of genus zero with exactly n + 1 cone points, each one of order p; in this case H is called a generalized Fermat group of type (p, n). In this case, it is known that S is non-hyperelliptic and that H is its unique generalized Fermat group of type (p, n). Also, explicit equations for them, as a fiber product of classical Fermat curves of degree p, are known. For p a prime integer, we describe those subgroups K of H acting freely on S, together with algebraic equations for S/K, and determine those K such that S/K is hyperelliptic. [ABSTRACT FROM AUTHOR]

We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in A × B × C , a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses roughly O (n 28 / 15) constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to many other problems where we seek a triple that satisfies a single polynomial equation, e.g., determining whether A × B × C contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. (2017) and by Chan (2018), where all three sets A, B, and C are assumed to be one-dimensional. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple (a , b , c) ∈ A × B × C that simultaneously satisfies two independent real polynomial equations. For example, this is the setup when testing for collinearity in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. We show that problems of this kind can be solved with roughly O (n 24 / 13) constant-degree polynomial sign tests. [ABSTRACT FROM AUTHOR]

The hull of a linear code C is the intersection of C with its dual C ⊥ . The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The hull of linear codes has recently found its application to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, we provide a new method to construct linear codes with one-dimensional hull. This construction method improves the code lengths and dimensions of the recent results given by the author. As a consequence, we derive several new classes of optimal linear codes with one-dimensional hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

The original algorithm of D.V. Chudnovsky and G.V. Chudnovsky for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear with respect to the degree of the extension. Recently, Randriambololona generalized the method, allowing asymmetry in the interpolation procedure. The aim of this article is to make effective this method. We first make explicit this generalization in order to construct the underlying asymmetric algorithms. Then, we propose a generic strategy to construct these algorithms using places of higher degrees and without derivated evaluation. Finally, we provide examples of three multiplication algorithms along with their Magma implementation: in F 16 13 using only rational places, in F 4 5 using also places of degree two, and in F 2 5 using also places of degree four. [ABSTRACT FROM AUTHOR]

We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then obtained by making corrections at singular points. We report on an implementation and provide some examples in MAGMA which demonstrate an improvement over Tuitman's algorithm. [ABSTRACT FROM AUTHOR]

We present a complete algorithm for computing the affine equivalences between two implicit planar algebraic curves. We provide evidence of the efficiency of the algorithm, implemented in Maple, and compare its performance with existing algorithms. As a part of the process for developing the algorithm, we characterize planar algebraic curves, possibly singular, possibly reducible, invariant under infinitely many affine equivalences. [ABSTRACT FROM AUTHOR]

We present the formula for the number of monic irreducible polynomials of degree n over the finite field F q where the coefficients of x n - 1 and x vanish for n ≥ 3 . In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements a ∈ F q n for which Trace (a) = 0 and Trace (a - 1) = 0 . [ABSTRACT FROM AUTHOR]

We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2, 3, 5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable over certain base fields. We give an example of a Burkhardt quartic over a bivariate function field whose desingularization has no rational points at all. Our methods are based on the representation theory of Sp 4 (F 3) , Galois cohomology, and the classical algebraic geometry of the Burkhardt quartic. [ABSTRACT FROM AUTHOR]

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method. [ABSTRACT FROM AUTHOR]

Let P ∈ Z [ X , Y ] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ , and let V R (P) : = { (x , y) ∈ R 2 ∣ P (x , y) = 0 } be the real planar algebraic curve implicitly defined as the vanishing set of P. We give a deterministic algorithm to compute the topology of V R (P) in terms of a simple straight-line planar graph G that is isotopic to V R (P) . The upper bound on the bit complexity of our algorithm is in O ~ (d 5 τ + d 6) (The expression "the complexity is in O ~ (f (d , τ)) " with f a polynomial in d , τ is an abbreviation for the expression "there exists a positive integer c such that the complexity is in O ((log d log τ) c f (d , τ)) "); which matches the current record bound for the problem of computing the topology of a planar algebraic curve. However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and more importantly the returned simple planar graph G yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points. [ABSTRACT FROM AUTHOR]

In this work, we construct, for any d ≥ 2 , a new foliation on CP 2 of degree d with a unique singular point of multiplicity d - 1 without invariant algebraic curves that contain all its separatrices. We also prove that if X is a foliation on CP 2 with a unique nilpotent singular point, then X has no algebraic leaves. Finally, we characterize logarithmic foliations on CP 2 with a unique singular point. And we give some new examples of this kind of foliations. [ABSTRACT FROM AUTHOR]

Given a set S of n points in R d , a k-set is a subset of k points of S that can be strictly separated by a hyperplane from the remaining n - k points. Similarly, one may consider k-facets, which are hyperplanes that pass through d points of S and have k points on one side. A notorious open problem is to determine the asymptotics of the maximum number of k-sets. In this paper we study a variation on the k-set/k-facet problem with hyperplanes replaced by algebraic surfaces. In stark contrast to the original k-set/k-facet problem, there are some natural families of algebraic curves for which the number of k-facets can be counted exactly. For example, we show that the number of halving conic sections for any set of 2 n + 5 points in general position in the plane is 2 n + 2 2 2 . To understand the limits of our argument we study a class of maps we call generally neighborly embeddings, which map generic point sets into neighborly position. Additionally, we give a simple argument which improves the best known bound on the number of k-sets/k-facets for point sets in convex position. [ABSTRACT FROM AUTHOR]

We construct analytically two algebraic closed curves forming a Poincaré–Bendixson annulus for the van der Pol system for all values of its parameter. The inner boundary of the annulus is a closed curve of the zero-level set of a Dulac–Cherkas function, which implies that this annulus contains at most one limit cycle. For the construction of the outer boundary a special procedure is presented. [ABSTRACT FROM AUTHOR]

Several non equivalent definitions exist for the envelope of a 1-parameter family of plane curves. Another notion, often considered as related to envelopes, is the offset at a given distance of a plane curve. Using the so-called analytic definition, we study and compare the envelope of a 1-parameter family of circles centered on a parabola and an offset of this parabola. Then we perform a similar study for an astroid. This illustrates the non equivalence of the two notions. The work is performed with networking with a Computer Algebra System and a Dynamical Geometry System, bringing them into a certain form of dialog. The obtained curves are sextics, and their discovery as envelopes leads to an automated exploration of Talbot curves. Another output shows these curves within a unifying framework involving a Maltese Cross. [ABSTRACT FROM AUTHOR]

Let X be a (projective, algebraic, non-singular, absolutely irreducible) curve of genus g defined over an algebraically closed field K of characteristic p ≥ 0 , and let q be a prime dividing the cardinality of Aut (X) . We say that X is a q-curve. Homma proved that either q ≤ g + 1 or q = 2 g + 1 , and classified (2 g + 1) -curves up to birational equivalence. In this note, we give the analogous classification for (g + 1) -curves, including a characterization of hyperelliptic (g + 1) -curves. Also, we provide the characterization of the full automorphism groups of q-curves for q = 2 g + 1 , g + 1 . Here, we make use of two different techniques: the former case is handled via a result by Vdovin bounding the size of abelian subgroups of finite simple groups, the second via the classification by Giulietti and Korchmáros of automorphism groups of curves of even genus. Finally, we give some partial results on the classification of q-curves for q = g , g - 1 . [ABSTRACT FROM AUTHOR]

Richard Guy asked the following question: can we find a triangle with rational sides, medians and area? Such a triangle is called a perfect triangle and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle 0 < θ < π , the number of perfect triangles with an angle θ is finite. [ABSTRACT FROM AUTHOR]

Algorithms and software for two new methods of solving polynomial equations based on constructing a convex polygon are described. The first method approximately solves the equations using the Hadamard polygon. The second method makes it possible to find branches of an algebraic curve in the vicinity of its singular points and in the vicinity of infinity using the Newton polygon and sketch real algebraic curves on the plane. The corresponding geometries and computer algebra algorithms for analyzing arbitrarily complicated cases are discussed. [ABSTRACT FROM AUTHOR]