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Journal articles on the topic 'Geometry, Algebraic'

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Published: 4 June 2021
Last updated: 29 July 2024

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1

Hacon, Christopher, Daniel Huybrechts, Yujiro Kawamata, and Bernd Siebert. "Algebraic Geometry." Oberwolfach Reports 12, no. 1 (2015): 783–836. http://dx.doi.org/10.4171/owr/2015/15.

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2

PLOTKIN, BORIS. "SOME RESULTS AND PROBLEMS RELATED TO UNIVERSAL ALGEBRAIC GEOMETRY." International Journal of Algebra and Computation 17, no. 05n06 (August 2007): 1133–64. http://dx.doi.org/10.1142/s0218196707003986.

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In universal algebraic geometry (UAG), some primary notions of classical algebraic geometry are applied to an arbitrary variety of algebras Θ and an arbitrary algebra H ∈ Θ. We consider an algebraic geometry in Θ over the distinguished algebra H and we also analyze H from the point of view of its geometric properties. This insight leads to a system of new notions and stimulates a number of new problems. They are new with respect to algebra, algebraic geometry and even with respect to the classical algebraic geometry. In our approach, there are two main aspects: the first one is a study of the algebra H and its geometric properties, while the second is focused on studying algebraic sets and algebraic varieties over a "good", particular algebra H. Considering the subject from the second standpoint, the main goal is to get forward as far as possible in a classification of algebraic sets over the given H. The first approach does not require such a classification which is itself an independent and extremely difficult task. We also consider some geometric relations between different H1 and H2 in Θ. The present paper should be viewed as a brief review of what has been done in universal algebraic geometry. We also give a list of unsolved problems for future work.
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3

Tyurin, N. A. "Algebraic Lagrangian geometry: three geometric observations." Izvestiya: Mathematics 69, no. 1 (February 28, 2005): 177–90. http://dx.doi.org/10.1070/im2005v069n01abeh000527.

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4

Voisin, Claire. "Algebraic Geometry versus Kähler geometry." Milan Journal of Mathematics 78, no. 1 (March 17, 2010): 85–116. http://dx.doi.org/10.1007/s00032-010-0113-8.

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5

Toën, Bertrand. "Derived algebraic geometry." EMS Surveys in Mathematical Sciences 1, no. 2 (2014): 153–245. http://dx.doi.org/10.4171/emss/4.

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6

Debarre, Olivier, David Eisenbud, Gavril Farkas, and Ravi Vakil. "Classical Algebraic Geometry." Oberwolfach Reports 18, no. 2 (August 24, 2022): 1519–77. http://dx.doi.org/10.4171/owr/2021/29.

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7

Darke, Ian, and M. Reid. "Undergraduate Algebraic Geometry." Mathematical Gazette 73, no. 466 (December 1989): 351. http://dx.doi.org/10.2307/3619332.

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8

Debarre, Olivier, David Eisenbud, Frank-Olaf Schreyer, and Ravi Vakil. "Classical Algebraic Geometry." Oberwolfach Reports 9, no. 2 (2012): 1845–93. http://dx.doi.org/10.4171/owr/2012/30.

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9

Catanese, Fabrizio, Christopher Hacon, Yujiro Kawamata, and Bernd Siebert. "Complex Algebraic Geometry." Oberwolfach Reports 10, no. 2 (2013): 1563–627. http://dx.doi.org/10.4171/owr/2013/27.

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10

Debarre, Olivier, David Eisenbud, Gavril Farkas, and Ravi Vakil. "Classical Algebraic Geometry." Oberwolfach Reports 11, no. 3 (2014): 1695–745. http://dx.doi.org/10.4171/owr/2014/31.

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11

Debarre, Olivier, David Eisenbud, Gavril Farkas, and Ravi Vakil. "Classical Algebraic Geometry." Oberwolfach Reports 13, no. 2 (2016): 1635–82. http://dx.doi.org/10.4171/owr/2016/29.

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12

Debarre, Olivier, David Eisenbud, Gavril Farkas, and Ravi Vakil. "Classical Algebraic Geometry." Oberwolfach Reports 15, no. 3 (August 26, 2019): 1983–2031. http://dx.doi.org/10.4171/owr/2018/33.

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13

Leykin, Anton. "Numerical algebraic geometry." Journal of Software for Algebra and Geometry 3, no. 1 (2011): 5–10. http://dx.doi.org/10.2140/jsag.2011.3.5.

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14

ODAGIRI, Shinsuke. "Tropical algebraic geometry." Hokkaido Mathematical Journal 38, no. 4 (November 2009): 771–95. http://dx.doi.org/10.14492/hokmj/1258554243.

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15

Blake, I., C. Heegard, T. Hoholdt, and V. Wei. "Algebraic-geometry codes." IEEE Transactions on Information Theory 44, no. 6 (1998): 2596–618. http://dx.doi.org/10.1109/18.720550.

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16

Shaska, T. "Computational algebraic geometry." Journal of Symbolic Computation 57 (October 2013): 1–2. http://dx.doi.org/10.1016/j.jsc.2013.05.001.

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17

Nishimura, Hirokazu. "Empirical algebraic geometry." International Journal of Theoretical Physics 34, no. 3 (March 1995): 305–21. http://dx.doi.org/10.1007/bf00671594.

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18

Joyce, D. "Hypercomplex Algebraic Geometry." Quarterly Journal of Mathematics 49, no. 2 (June 1, 1998): 129–62. http://dx.doi.org/10.1093/qmathj/49.2.129.

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19

Daniyarova, E. Yu, A. G. Myasnikov, and V. N. Remeslennikov. "Universal algebraic geometry." Doklady Mathematics 84, no. 1 (August 2011): 545–47. http://dx.doi.org/10.1134/s1064562411050073.

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20

Verschelde, J. "Basic algebraic geometry." Journal of Computational and Applied Mathematics 66, no. 1-2 (January 1996): N3—N4. http://dx.doi.org/10.1016/0377-0427(96)80471-7.

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21

Wampler, Charles W., and Andrew J. Sommese. "Numerical algebraic geometry and algebraic kinematics." Acta Numerica 20 (April 28, 2011): 469–567. http://dx.doi.org/10.1017/s0962492911000067.

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In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.
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22

Bambozzi, Federico, and Oren Ben-Bassat. "Dagger geometry as Banach algebraic geometry." Journal of Number Theory 162 (May 2016): 391–462. http://dx.doi.org/10.1016/j.jnt.2015.10.023.

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23

Sederberg, Thomas, and Ronald Goldman. "Algebraic Geometry for Computer-Aided Geometric Design." IEEE Computer Graphics and Applications 6, no. 6 (1986): 52–59. http://dx.doi.org/10.1109/mcg.1986.276742.

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24

Daniyarova, E. Yu, A. G. Myasnikov, and V. N. Remeslennikov. "Algebraic geometry over algebraic structures. II. Foundations." Journal of Mathematical Sciences 185, no. 3 (August 1, 2012): 389–416. http://dx.doi.org/10.1007/s10958-012-0923-z.

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25

BELLUCE, LAWRENCE P., ANTONIO DI NOLA, and GIACOMO LENZI. "ALGEBRAIC GEOMETRY FOR MV-ALGEBRAS." Journal of Symbolic Logic 79, no. 4 (December 2014): 1061–91. http://dx.doi.org/10.1017/jsl.2014.53.

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AbstractIn this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.
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26

Chandrashekara A C. "System Hypothesis Implications of Algebraic Geometry." international journal of engineering technology and management sciences 7, no. 1 (February 28, 2023): 105–8. http://dx.doi.org/10.46647/ijetms.2023.v07i01.018.

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Certain pole-placement concepts, such as an enhanced form of pole location with output response, are proven using fundamental algebraic geometry equations. Illustrations that highlight the algebra-geometric equations drawbacks and its possible application to systems analysis are shown. This study and ones that may come after it may help to make the potent theorems of current algebraic geometry comprehensible and useful for solving technical hurdles.
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27

Totaro, Burt. "Euler and algebraic geometry." Bulletin of the American Mathematical Society 44, no. 04 (June 22, 2007): 541–60. http://dx.doi.org/10.1090/s0273-0979-07-01178-0.

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28

Jin, Lingfei, Yuan Luo, and Chaoping Xing. "Repairing Algebraic Geometry Codes." IEEE Transactions on Information Theory 64, no. 2 (February 2018): 900–908. http://dx.doi.org/10.1109/tit.2017.2773089.

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29

Bashmakova, I. G., and E. I. Slavutin. "Glimpses of Algebraic Geometry." American Mathematical Monthly 104, no. 1 (January 1997): 62. http://dx.doi.org/10.2307/2974826.

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30

Gorodentsev, A. L., and A. N. Tyurin. "Abelian Lagrangian algebraic geometry." Izvestiya: Mathematics 65, no. 3 (June 30, 2001): 437–67. http://dx.doi.org/10.1070/im2001v065n03abeh000334.

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31

Sergeraert, F. "Geometry and algebraic topology." Mathematics and Computers in Simulation 42, no. 4-6 (November 1996): 459. http://dx.doi.org/10.1016/s0378-4754(96)00020-1.

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32

Winkler, Franz. "Algebraic computation in geometry." Mathematics and Computers in Simulation 42, no. 4-6 (November 1996): 529–37. http://dx.doi.org/10.1016/s0378-4754(96)00028-6.

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33

Rota, Gian-Carlo. "Geometry of algebraic curves." Advances in Mathematics 80, no. 1 (March 1990): 134. http://dx.doi.org/10.1016/0001-8708(90)90021-e.

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34

Jun, Jaiung. "Algebraic geometry over hyperrings." Advances in Mathematics 323 (January 2018): 142–92. http://dx.doi.org/10.1016/j.aim.2017.10.043.

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35

Bashmakova, I. G., and E. I. Slavutin. "Glimpses of Algebraic Geometry." American Mathematical Monthly 104, no. 1 (January 1997): 62–67. http://dx.doi.org/10.1080/00029890.1997.11990599.

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36

Gabber, Ofer, and Shane Kelly. "Points in algebraic geometry." Journal of Pure and Applied Algebra 219, no. 10 (October 2015): 4667–80. http://dx.doi.org/10.1016/j.jpaa.2015.03.001.

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37

Grandau, Laura, and Ana C. Stephens. "Algebraic Thinking and Geometry." Mathematics Teaching in the Middle School 11, no. 7 (March 2006): 344–49. http://dx.doi.org/10.5951/mtms.11.7.0344.

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Research on the learning and teaching of algebra has recently been identified as a priority by members of the mathematics education research community (e.g., Ball 2003; Carpenter and Levi 2000; Kaput 1998; Olive, Izsak, and Blanton 2002). Rather than view algebra as an isolated course of study to be completed in the eighth or ninth grade, these researchers advocate the reconceptualization of algebra as a strand that weaves throughout other areas of mathematics in the K–12 curriculum.
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38

Watase, Yasushige. "Introduction to Algebraic Geometry." Formalized Mathematics 31, no. 1 (September 1, 2023): 67–73. http://dx.doi.org/10.2478/forma-2023-0007.

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Summary A classical algebraic geometry is study of zero points of system of multivariate polynomials [3], [7] and those zero points would be corresponding to points, lines, curves, surfaces in an affine space. In this article we give some basic definition of the area of affine algebraic geometry such as algebraic set, ideal of set of points, and those properties according to [4] in the Mizar system[5], [2]. We treat an affine space as the n-fold Cartesian product k n as the same manner appeared in [4]. Points in this space are identified as n-tuples of elements from the set k. The formalization of points, which are n-tuples of numbers, is described in terms of a mapping from n to k, where the domain n corresponds to the set n = {0, 1, . . ., n − 1}, and the target domain k is the same as the scalar ring or field of polynomials. The same approach has been applied when evaluating multivariate polynomials using n-tuples of numbers [10]. This formalization aims at providing basic notions of the field which enable to formalize geometric objects such as algebraic curves which is used e.g. in coding theory [11] as well as further formalization of the fields [8] in the Mizar system, including the theory of polynomials [6].
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39

Newelski, Ludomir. "Geometry of *-Finite Types." Journal of Symbolic Logic 64, no. 4 (December 1999): 1375–95. http://dx.doi.org/10.2307/2586784.

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AbstractAssume T is a superstable theory with < 2ℵ0 countable models. We prove that any *- algebraic type of -rank > 0 is m-nonorthogonal to a *-algebraic type of -rank 1. We study the geometry induced by m-dependence on a *-algebraic type p* of -rank 1. We prove that after some localization this geometry becomes projective over a division ring . Associated with p* is a meager type p. We prove that p is determined by p* up to nonorthogonality and that underlies also the geometry induced by forking dependence on any stationarization of p. Also we study some *-algebraic *-groups of -rank 1 and prove that any *-algebraic *-group of -rank 1 is abelian-by-finite.
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40

Palacios, Joe, and Ruth Cabanillas. "Computational methods in algebraic geometry." Selecciones Matemáticas 10, no. 02 (November 30, 2023): 462–69. http://dx.doi.org/10.17268/sel.mat.2023.02.17.

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In this article we give a general description of the field of algebraic geometry and its computational aspects using Macaulay 2. In particular, we present an application to the bidimensional robotic arm of n spans for any natural n.
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41

Toën, Bertrand, and Gabriele Vezzosi. "Homotopical algebraic geometry. II. Geometric stacks and applications." Memoirs of the American Mathematical Society 193, no. 902 (2008): 0. http://dx.doi.org/10.1090/memo/0902.

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42

Biswas, Indranil, and Sorin Dumitrescu. "Fujiki class 𝒞 and holomorphic geometric structures." International Journal of Mathematics 31, no. 05 (April 9, 2020): 2050039. http://dx.doi.org/10.1142/s0129167x20500391.

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For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class [Formula: see text], whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class [Formula: see text] and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.
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43

Daniyarova, Evelina Yur’evna, Alexei Georgievich Myasnikov, and Vladimir Nikanorovich Remeslennikov. "Algebraic geometry over algebraic structures X: Ordinal dimension." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1425–48. http://dx.doi.org/10.1142/s0218196718400039.

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This work is devoted to interpretation of concepts of Zariski dimension of an algebraic variety over a field and of Krull dimension of a coordinate ring in algebraic geometry over algebraic structures of an arbitrary signature. Proposed dimensions are ordinal numbers (ordinals).
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44

Daniyarova, E. Yu, A. G. Myasnikov, and V. N. Remeslennikov. "Algebraic Geometry Over Algebraic Structures. VI. Geometrical Equivalence." Algebra and Logic 56, no. 4 (September 2017): 281–94. http://dx.doi.org/10.1007/s10469-017-9449-2.

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45

Baldwin, John T., and Andreas Mueller. "Autonomy of Geometry." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 11 (February 5, 2020): 5–24. http://dx.doi.org/10.24917/20809751.11.1.

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In this paper we present three aspects of the autonomy of geometry. (1) An argument for the geometric as opposed to the ‘geometric algebraic’ interpretation of Euclid’s Books I and II; (2) Hilbert’s successful project to axiomatize Euclid’s geometry in a first order geometric language, notably eliminating the dependence on the Archimedean axiom; (3) the independent conception of multiplication from a geometric as opposed to an arithmetic viewpoint.
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46

MARTIN, MIRCEA. "ALGEBRA ENVIRONMENTS I." Revue Roumaine Mathematiques Pures Appliquees LXIX, no. 1 (April 1, 2024): 17–60. http://dx.doi.org/10.59277/rrmpa.2024.17.60.

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Algebra environments capture properties of non–commutative conditional expectations in a general algebraic setting. Their study relies on algebraic geometry, topology, and differential geometry techniques. The structure algebraic and Banach manifolds of algebra environments and their Zariski and smooth tangent vector bundles are particular objects of interest. A description of derivations on algebra environments compatible with geometric structures is an additional issue. Grassmann and flag manifolds of unital involutive algebras and spaces of projective compact group representations in C∗–algebras are analyzed as structure manifolds of associated algebra environments.
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47

Kimuya, Alex M., and Stephen Mbugua Karanja. "Incompatibility between Euclidean Geometry and the Algebraic Solutions of Geometric Problems." European Journal of Mathematics and Statistics 4, no. 4 (July 10, 2023): 14–23. http://dx.doi.org/10.24018/ejmath.2023.4.4.90.

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The transition from the “early-modern” mathematical and scientific norms of establishing conventional Euclidean geometric proofs has experienced quite mixed modes of reasoning. For instance, a careful investigation based on the continued attempts by different practitioners to resolve the geometric trisectability of a plane angle suggests serious hitches with the established algebraic angles non-trisectability proofs. These faults found the root for the difficult geometric question about having straightedge and compass proofs for either the trisectability or the non-trisectability of angles. One of the evident gaps regarding the norms for establishing the Euclidean geometric proofs concerns the incompatibility between the smugly asserted algebraic-geometric proofs and the desired inherent Euclidean geometric proofs. We consider an algebraically translated proof of the geometric angle trisection scheme proposed by [1]. We assert and prove that there is a complete incompatibility between the geometric and the algebraic methods of proofs, and hence the algebraic methods should not be used as authoritative means of proving Euclidean geometric problems. The paper culminates by employing the incompatibility proofs in justifying the independence of the Euclidean geometric system.
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48

Iwanari, Isamu. "The category of toric stacks." Compositio Mathematica 145, no. 03 (May 2009): 718–46. http://dx.doi.org/10.1112/s0010437x09003911.

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AbstractIn this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.
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49

Ito, Atsushi, Makoto Miura, and Kazushi Ueda. "Projective Reconstruction in Algebraic Vision." Canadian Mathematical Bulletin 63, no. 3 (November 13, 2019): 592–609. http://dx.doi.org/10.4153/s0008439519000687.

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AbstractWe discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.
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50

Ben-Bassat, Oren, and Kobi Kremnizer. "Non-Archimedean analytic geometry as relative algebraic geometry." Annales de la faculté des sciences de Toulouse Mathématiques 26, no. 1 (2017): 49–126. http://dx.doi.org/10.5802/afst.1526.

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