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Journal articles on the topic 'Semialgebraic and subanalytic geometry'

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Author: Grafiati

Published: 1 September 2023

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1

Coste, Michel. "Book Review: Geometry of subanalytic and semialgebraic sets." Bulletin of the American Mathematical Society 36, no. 04 (July 27, 1999): 523–28. http://dx.doi.org/10.1090/s0273-0979-99-00793-4.

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2

Loi, Ta Lê. "Transversality theorem in o-minimal structures." Compositio Mathematica 144, no. 5 (September 2008): 1227–34. http://dx.doi.org/10.1112/s0010437x08003503.

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AbstractIn this paper we present Thom’s transversality theorem in o-minimal structures (a generalization of semialgebraic and subanalytic geometry). There are no restrictions on the differentiability class and the dimensions of manifolds involved in comparison withthe general case.

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3

Figueiredo, Rodrigo. "O-minimal de Rham Cohomology." Bulletin of Symbolic Logic 28, no. 4 (December 2022): 529. http://dx.doi.org/10.1017/bsl.2021.20.

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AbstractO-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.Abstract prepared by Rodrigo Figueiredo.E-mail: rodrigo@ime.usp.brURL: https://doi.org/10.11606/T.45.2019.tde-28042019-181150

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4

KOVACSICS, PABLO CUBIDES, and KIEN HUU NGUYEN. "A P-MINIMAL STRUCTURE WITHOUT DEFINABLE SKOLEM FUNCTIONS." Journal of Symbolic Logic 82, no. 2 (May 15, 2017): 778–86. http://dx.doi.org/10.1017/jsl.2016.58.

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AbstractWe show there are intermediate P-minimal structures between the semialgebraic and subanalytic languages which do not have definable Skolem functions. As a consequence, by a result of Mourgues, this shows there are P-minimal structures which do not admit classical cell decomposition.

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5

Kaiser, Tobias. "Capacity in subanalytic geometry." Illinois Journal of Mathematics 49, no. 3 (July 2005): 719–36. http://dx.doi.org/10.1215/ijm/1258138216.

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6

Niederman, Laurent. "Hamiltonian stability and subanalytic geometry." Annales de l’institut Fourier 56, no. 3 (2006): 795–813. http://dx.doi.org/10.5802/aif.2200.

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7

Zeng, Guangxin. "Homogeneous Stellensätze in semialgebraic geometry." Pacific Journal of Mathematics 136, no. 1 (January 1, 1989): 103–22. http://dx.doi.org/10.2140/pjm.1989.136.103.

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8

Łojasiewicz, Stanisław. "On semi-analytic and subanalytic geometry." Banach Center Publications 34, no. 1 (1995): 89–104. http://dx.doi.org/10.4064/-34-1-89-104.

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9

Qi, Yang, Pierre Comon, and Lek-Heng Lim. "Semialgebraic Geometry of Nonnegative Tensor Rank." SIAM Journal on Matrix Analysis and Applications 37, no. 4 (January 2016): 1556–80. http://dx.doi.org/10.1137/16m1063708.

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10

Solernó, Pablo. "Effective Łojasiewicz inequalities in semialgebraic geometry." Applicable Algebra in Engineering, Communication and Computing 2, no. 1 (March 1991): 1–14. http://dx.doi.org/10.1007/bf01810850.

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11

Choi, Myung-Jun, Tomohiro Kawakami, and Dae Heui Park. "Equivariant semialgebraic vector bundles." Topology and its Applications 123, no. 2 (September 2002): 383–400. http://dx.doi.org/10.1016/s0166-8641(01)00291-7.

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12

Choi, Myung-Jun, Dae Heui Park, and Dong Youp Suh. "Proof of semialgebraic covering mapping cylinder conjecture with semialgebraic covering homotopy theorem." Topology and its Applications 154, no. 1 (January 2007): 69–89. http://dx.doi.org/10.1016/j.topol.2006.03.017.

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13

Valette, Anna, and Guillaume Valette. "Poincaré Inequality on Subanalytic Sets." Journal of Geometric Analysis 31, no. 10 (March 26, 2021): 10464–72. http://dx.doi.org/10.1007/s12220-021-00652-x.

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AbstractLet $$\Omega $$ Ω be a subanalytic connected bounded open subset of $$\mathbb {R}^n$$ R n , with possibly singular boundary. We show that given $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) , there is a constant C such that for any $$u\in W^{1,p}(\Omega )$$ u ∈ W 1 , p ( Ω ) we have $$||u-u_{\Omega }||_{L^p} \le C||\nabla u||_{L^p},$$ | | u - u Ω | | L p ≤ C | | ∇ u | | L p , where we have set $$u_{\Omega }:=\frac{1}{|\Omega |}\int _{\Omega } u.$$ u Ω : = 1 | Ω | ∫ Ω u .

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14

Lasserre, Jean B. "Convexity in SemiAlgebraic Geometry and Polynomial Optimization." SIAM Journal on Optimization 19, no. 4 (January 2009): 1995–2014. http://dx.doi.org/10.1137/080728214.

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15

Cimprič, J. "Archimedean preorderings in non-commutative semialgebraic geometry." Communications in Algebra 28, no. 3 (January 2000): 1603–14. http://dx.doi.org/10.1080/00927870008826916.

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16

Gardener, T. S., and Hans Schoutens. "Flattening and Subanalytic Sets in Rigid Analytic Geometry." Proceedings of the London Mathematical Society 83, no. 3 (November 2001): 681–707. http://dx.doi.org/10.1112/plms/83.3.681.

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17

Kankaanrinta, Marja. "Equivariant uniformization theorem for subanalytic sets." Geometriae Dedicata 136, no. 1 (August 22, 2008): 167–73. http://dx.doi.org/10.1007/s10711-008-9282-9.

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18

Acquistapace, F., R. Benedetti, and F. Broglia. "Effectiveness?non effectiveness in semialgebraic and PL geometry." Inventiones Mathematicae 102, no. 1 (December 1990): 141–56. http://dx.doi.org/10.1007/bf01233424.

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19

FERNANDO, JOSÉ F., and J. M. GAMBOA. "ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET." International Journal of Mathematics 23, no. 04 (April 2012): 1250031. http://dx.doi.org/10.1142/s0129167x12500310.

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In this work we define a semialgebraic set S ⊂ ℝn to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring [Formula: see text]: Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

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20

Tannenbaum, A., and Y. Yomdin. "Robotic manipulators and the geometry of real semialgebraic sets." IEEE Journal on Robotics and Automation 3, no. 4 (August 1987): 301–7. http://dx.doi.org/10.1109/jra.1987.1087105.

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21

Huber, Roland, and Claus Scheiderer. "A relative notion of local completeness in semialgebraic geometry." Archiv der Mathematik 53, no. 6 (December 1989): 571–84. http://dx.doi.org/10.1007/bf01199817.

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22

Cucker, Felipe. "Grid Methods in Computational Real Algebraic (and Semialgebraic) Geometry." Chinese Annals of Mathematics, Series B 39, no. 2 (February 2, 2018): 373–96. http://dx.doi.org/10.1007/s11401-018-1070-8.

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23

Pierzchała, Rafał. "An estimate for the Siciak extremal function – Subanalytic geometry approach." Journal of Mathematical Analysis and Applications 430, no. 2 (October 2015): 755–76. http://dx.doi.org/10.1016/j.jmaa.2015.05.012.

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24

Nowak, Krzysztof Jan. "Decomposition into special cubes and its applications to quasi-subanalytic geometry." Annales Polonici Mathematici 96, no. 1 (2009): 65–74. http://dx.doi.org/10.4064/ap96-1-6.

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25

Nicolaescu, Liviu I. "On the normal cycles of subanalytic sets." Annals of Global Analysis and Geometry 39, no. 4 (December 16, 2010): 427–54. http://dx.doi.org/10.1007/s10455-010-9241-1.

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26

Bos, L. P., and P. D. Milman. "Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains." Geometric and Functional Analysis 5, no. 6 (November 1995): 853–923. http://dx.doi.org/10.1007/bf01902214.

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27

Cluckers, Raf, and Immanuel Halupczok. "Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry." Selecta Mathematica 18, no. 4 (February 17, 2012): 825–37. http://dx.doi.org/10.1007/s00029-012-0088-0.

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28

Comte, Georges, and Goulwen Fichou. "Grothendieck ring of semialgebraic formulas and motivic real Milnor fibers." Geometry & Topology 18, no. 2 (April 7, 2014): 963–96. http://dx.doi.org/10.2140/gt.2014.18.963.

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29

Kurdyka, Krzysztof, Stanisław Spodzieja, and Anna Szlachcińska. "Metric Properties of Semialgebraic Mappings." Discrete & Computational Geometry 55, no. 4 (March 21, 2016): 786–800. http://dx.doi.org/10.1007/s00454-016-9776-4.

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30

Kankaanrinta, Marja. "The simplicity of certain groups of subanalytic diffeomorphisms." Differential Geometry and its Applications 27, no. 5 (October 2009): 661–70. http://dx.doi.org/10.1016/j.difgeo.2009.03.006.

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31

Cluckers, Raf, Georges Comte, and François Loeser. "Lipschitz Continuity Properties for p−Adic Semi-Algebraic and Subanalytic Functions." Geometric and Functional Analysis 20, no. 1 (April 23, 2010): 68–87. http://dx.doi.org/10.1007/s00039-010-0060-0.

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32

Kahle, Thomas, Frank Röttger, and Rainer Schwabe. "The semialgebraic geometry of saturated optimal designs for the Bradley–Terry model." Algebraic Statistics 12, no. 1 (April 9, 2021): 97–114. http://dx.doi.org/10.2140/astat.2021.12.97.

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33

Becker, Eberhard. "On the real spectrum of a ring and its application to semialgebraic geometry." Bulletin of the American Mathematical Society 15, no. 1 (July 1, 1986): 19–61. http://dx.doi.org/10.1090/s0273-0979-1986-15431-5.

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34

Kurdyka, Krzysztof, Stanisław Spodzieja, and Anna Szlachcińska. "Correction to: Metric Properties of Semialgebraic Mappings." Discrete & Computational Geometry 62, no. 4 (September 3, 2019): 990–91. http://dx.doi.org/10.1007/s00454-019-00128-4.

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35

Fox, Jacob, János Pach, and Andrew Suk. "A Polynomial Regularity Lemma for Semialgebraic Hypergraphs and Its Applications in Geometry and Property Testing." SIAM Journal on Computing 45, no. 6 (January 2016): 2199–223. http://dx.doi.org/10.1137/15m1007355.

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36

Gabrielov, Andrei, and Nicolai Vorobjov. "Betti Numbers of Semialgebraic Sets Defined by Quantifier-Free Formulae." Discrete & Computational Geometry 33, no. 3 (May 28, 2004): 395–401. http://dx.doi.org/10.1007/s00454-004-1105-7.

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37

Dutta, Kunal, Arijit Ghosh, Bruno Jartoux, and Nabil H. Mustafa. "Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning." Discrete & Computational Geometry 61, no. 4 (March 15, 2019): 756–77. http://dx.doi.org/10.1007/s00454-019-00075-0.

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38

Nowak, Krzysztof Jan. "Addendum to the paper ``Decomposition into special cubes and its application to quasi-subanalytic geometry'' (Ann. Polon. Math. 96 (2009), 65–74)." Annales Polonici Mathematici 98, no. 2 (2010): 201–5. http://dx.doi.org/10.4064/ap98-2-7.

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39

Chieu, N. H., J. W. Feng, W. Gao, G. Li, and D. Wu. "SOS-Convex Semialgebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization." Set-Valued and Variational Analysis 26, no. 2 (November 14, 2017): 305–26. http://dx.doi.org/10.1007/s11228-017-0456-1.

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40

Bürgisser, Peter, and Antonio Lerario. "Probabilistic Schubert calculus." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 760 (March 1, 2020): 1–58. http://dx.doi.org/10.1515/crelle-2018-0009.

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AbstractWe initiate the study of average intersection theory in real Grassmannians. We define the expected degree{\operatorname{edeg}G(k,n)} of the real Grassmannian {G(k,n)} as the average number of real k-planes meeting nontrivially {k(n-k)} random subspaces of {\mathbb{R}^{n}}, all of dimension {n-k}, where these subspaces are sampled uniformly and independently from {G(n-k,n)}. We express {\operatorname{edeg}G(k,n)} in terms of the volume of an invariant convex body in the tangent space to the Grassmannian, and prove that for fixed {k\geq 2} and {n\to\infty},\operatorname{edeg}G(k,n)=\deg G_{\mathbb{C}}(k,n)^{\frac{1}{2}\varepsilon_{k}% +o(1)},where {\deg G_{\mathbb{C}}(k,n)} denotes the degree of the corresponding complex Grassmannian and {\varepsilon_{k}} is monotonically decreasing with {\lim_{k\to\infty}\varepsilon_{k}=1}. In the case of the Grassmannian of lines, we prove the finer asymptotic\operatorname{edeg}G(2,n+1)=\frac{8}{3\pi^{5/2}\sqrt{n}}\biggl{(}\frac{\pi^{2}% }{4}\biggr{)}^{n}(1+\mathcal{O}(n^{-1})).The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set {X\subseteq\mathbb{R}\mathrm{P}^{n-1}} of dimension {n-k-1} its Chow hypersurface {Z(X)\subseteq G(k,n)}, consisting of the k-planes A in {\mathbb{R}^{n}} whose projectivization intersects X. Denoting {N:=k(n-k)}, we show that\mathbb{E}\#(g_{1}Z(X_{1})\cap\cdots\cap g_{N}Z(X_{N}))=\operatorname{edeg}G(k% ,n)\cdot\prod_{i=1}^{N}\frac{|X_{i}|}{|\mathbb{R}\mathrm{P}^{m}|},where each {X_{i}} is of dimension {m=n-k-1}, the expectation is taken with respect to independent uniformly distributed {g_{1},\dots,g_{m}\in O(n)} and {|X_{i}|} denotes the m-dimensional volume of {X_{i}}.

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41

Sampaio, José Edson. "Globally subanalytic CMC surfaces in ℝ3 with singularities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics, March 30, 2020, 1–18. http://dx.doi.org/10.1017/prm.2020.21.

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Abstract In this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.

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42

Birbrair, Lev, Michael Brandenbursky, and Andrei Gabrielov. "Lipschitz geometry of surface germs in $${\mathbb {R}}^4$$: metric knots." Selecta Mathematica 29, no. 3 (May 19, 2023). http://dx.doi.org/10.1007/s00029-023-00847-w.

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AbstractA link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $${\mathbb {R}}^4$$ R 4 is a topological knot (or link) in $$S^3$$ S 3 . We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $${\mathbb {R}}^4$$ R 4 and knot theory. Namely, for any knot K, we construct a surface $$X_K$$ X K in $${\mathbb {R}}^4$$ R 4 such that: the link at the origin of $$X_{K}$$ X K is a trivial knot; the germs $$X_K$$ X K are outer bi-Lipschitz equivalent for all K; two germs $$X_{K}$$ X K and $$X_{K'}$$ X K ′ are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and $$K'$$ K ′ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in $${\mathbb {R}}^4$$ R 4 , even when they are topologically trivial and outer bi-Lipschitz equivalent.

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43

Drescher, Tom, Tim Netzer, and Andreas Thom. "On projections of free semialgebraic sets." Advances in Geometry, January 31, 2023. http://dx.doi.org/10.1515/advgeom-2022-0021.

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Abstract We examine to what extent a projection theorem is possible in the non-commutative (free) setting. We first review and extend some results that count against a full free projection theorem. We then obtain a weak version of the projection theorem: projections along linear and separated variables yield semialgebraically parametrised free semi-algebraic sets.

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44

Koike, Satoshi, and Laurentiu Paunescu. "Stabilisation of geometric directional bundle for a subanalytic set." Topology and its Applications, December 2021, 107988. http://dx.doi.org/10.1016/j.topol.2021.107988.

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45

Birbrair, Lev, and Andrei Gabrielov. "Lipschitz geometry of pairs of normally embedded Hölder triangles." European Journal of Mathematics, August 26, 2022. http://dx.doi.org/10.1007/s40879-022-00572-2.

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AbstractWe consider a special case of the outer bi-Lipschitz classification of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) surface germs, obtained as a union of two normally embedded Hölder triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called $$\sigma \tau $$ σ τ -pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.

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46

Basu, Saugata, and Negin Karisani. "Efficient simplicial replacement of semialgebraic sets." Forum of Mathematics, Sigma 11 (2023). http://dx.doi.org/10.1017/fms.2023.36.

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Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ , is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$ , there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ is $\ell $ -equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$ , where s is the number of polynomials appearing in the formula $\phi $ , and d a bound on their degrees. For fixed $\ell $ , this bound is singly exponential in k. In particular, since $\ell $ -equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$ .

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47

Averkov, Gennadiy, and Ludwig Bröcker. "Minimal polynomial descriptions of polyhedra and special semialgebraic sets." Advances in Geometry, January 7, 2012, 1–13. http://dx.doi.org/10.1515/advgeom.2011.059.

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48

Berlow, Katalin, Marie-Charlotte Brandenburg, Chiara Meroni, and Isabelle Shankar. "Intersection bodies of polytopes." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, January 25, 2022. http://dx.doi.org/10.1007/s13366-022-00621-7.

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AbstractWe investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for its computation. Moreover, we compute the irreducible components of the algebraic boundary and provide an upper bound for the degree of these components.

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49

De Pauw, Thierry, and Robert Hardt. "Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets." Journal of Singularities 24 (2022). http://dx.doi.org/10.5427/jsing.2022.24f.

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50

Cucker, Felipe, Alperen A. Ergür, and Josué Tonelli-Cueto. "Functional norms, condition numbers and numerical algorithms in algebraic geometry." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.89.

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Abstract In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand to optimise accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty }$ . This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty }$ -norm, the use of $L_p$ -norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale’s 17th problem).

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