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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

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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3

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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4

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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Анотація:
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.
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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Анотація:
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.
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

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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Анотація:
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.
26

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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Анотація:
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.
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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Анотація:
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.
38

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.
39

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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40

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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Анотація:
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).
41

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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42

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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Анотація:
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.
43

Biswas, Indranil, та Sorin Dumitrescu. "Fujiki class 𝒞 and holomorphic geometric structures". International Journal of Mathematics 31, № 05 (9 квітня 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.
44

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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45

Belavin, A. A., and V. G. Knizhnik. "Algebraic geometry and the geometry of quantum strings." Physics Letters B 168, no. 3 (March 1986): 201–6. http://dx.doi.org/10.1016/0370-2693(86)90963-9.

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46

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.
47

KOSHKİN, Sergiy. "Algebraic geometry on imaginary triangles." International Electronic Journal of Geometry 11, no. 2 (November 30, 2018): 71–82. http://dx.doi.org/10.36890/iejg.545133.

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48

Miranda, Rick. "Triple Covers in Algebraic Geometry." American Journal of Mathematics 107, no. 5 (October 1985): 1123. http://dx.doi.org/10.2307/2374349.

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49

Tyurin, N. A. "Lagrangian Geometry of Algebraic Manifolds." Physics of Particles and Nuclei Letters 19, no. 4 (July 26, 2022): 337–42. http://dx.doi.org/10.1134/s1547477122040215.

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50

Cabaña, Gustavo, María Chara, Ricardo Podestá, and Ricardo Toledano. "On cyclic algebraic-geometry codes." Finite Fields and Their Applications 82 (September 2022): 102064. http://dx.doi.org/10.1016/j.ffa.2022.102064.

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