Publication: On polynomial images of a closed ball
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In this work we approach the problem of determining which (compact) semialgebraic subsets of Rn are images under polynomial maps f : Rm → Rn of the closed unit ball Bm centered at the origin of some Euclidean space Rm and that of estimating (when possible) which is the smallest m with this property. Contrary to what happens with the images of Rm under polynomial maps, it is quite straightforward to provide basic examples of semialgebraic sets that are polynomial images of the closed unit ball. For instance, simplices, cylinders, hypercubes, elliptic, parabolic or hyperbolic segments (of dimension n) are polynomial images of the closedunit ball in Rn. The previous examples (and other basic ones proposed in the article) provide a large family of ‘n-bricks’ and we find necessary and sufficient conditions to guarantee that a finite union of ‘n-bricks’ is again a polynomial image of the closed unit ball either of dimension n or n + 1. In this direction, we prove: A finite union S of n-dimensional convex polyhedra is the image of the n-dimensional closed unit ball Bn if and only if S is connected by analytic paths. The previous result can be generalized using the ‘n-bricks’ mentioned before and we show: If S1, . . . , S` ⊂ Rn are ‘n-bricks’, the union S := S` i=1 Si is the image of the closed unit ball Bn+1 of Rn+1 under a polynomial map f: Rn+1 → Rn if and only if S is connected by analytic paths.
[BPR] S. Basu, R. Pollack, M.-F. Roy: Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10. Springer-Verlag, Berlin (2006). [Be] M. Berger: Geometry. I & II. Universitext. Springer-Verlag, Berlin (1987). [BCR] J. Bochnak, M. Coste, M.-F. Roy: Real algebraic geometry. Ergeb. Math. 36, Springer-Verlag, Berlin (1998). [B] I. Bogdanov: Approximation by polynomials, Mathoverflow, March 13(2012). https://mathoverflow.net/questions/91116/approximation-by-polynomials/91130#91130 [CF] A. Carbone, J.F. Fernando: Nash images of the closed unit ball. Preprint RAAG (2022). [E] D. Eisenbud: Open Problems in Computational Algebraic Geometry and commutative Algebra, in Computational Algebraic Geometry and Commutative Algebra, Cortona (1991), (ed. D. Eisenbud and L. Robbiano) Cambridge University Press, Cambridge, England, (1993), 49–71. [Fe1] J.F. Fernando: On the one dimensional polynomial and regular images of Rn. J. Pure Appl. Algebra 218(2014), no.9, 1745–1753. [Fe2] J.F. Fernando: On Nash images of Euclidean spaces. Adv. Math. 331 (2018), 627–719. [Fe3] J.F. Fernando: Polynomial and Nash paths inside semialgebraic sets. Preprint RAAG (2021). [FFQU] J.F. Fernando, G. Fichou, R. Quarez, C. Ueno: On regulous and regular images of Euclidean spaces, Q. J. Math. 69 (2018), no. 4, 1327–1351. [FG1] J.F. Fernando, J.M. Gamboa: Polynomial images of Rn. J. Pure Appl. Algebra 179 (2003), no. 3, 241–254. [FG2] J.F. Fernando, J.M. Gamboa: Polynomial and regular images of Rn. Israel J. Math. 153 (2006), 61–92. [FGU1] J.F. Fernando, J.M. Gamboa, C. Ueno: On convex polyhedra as regular images of R n . Proc. London Math. Soc. (3) 103 (2011), 847–878. [FGU2] J.F. Fernando, J.M. Gamboa, C. Ueno: The open quadrant problem: a topological proof. A mathematical tribute to Professor Jos´e Mar´ıa Montesinos Amilibia, 337–350, Dep. Geom. Topol. Fac. Cien. Mat. UCM, Madrid (2016). [FGU3] J.F. Fernando, J.M. Gamboa, C. Ueno: Polynomial, regular and Nash images of Euclidean spaces. Ordered algebraic structures and related topics, 145–167, Contemp. Math., 697, Amer. Math. Soc., Providence, RI (2017). [FGU4] J.F. Fernando, J.M. Gamboa, C. Ueno: Unbounded convex polyhedra as polynomial images of Euclidean spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 2, 509–565. [FU1] J.F. Fernando, C. Ueno: On the set of points at infinity of a polynomial image of Rn. Discrete Comput. Geom. 52 (2014), no. 4, 583–611. [FU2] J.F. Fernando, C. Ueno: On complements of convex polyhedra as polynomial and regular images of Rn. Int. Math. Res. Not. IMRN 2014, no. 18, 5084–5123. [FU3] J.F. Fernando, C. Ueno: On the complements of 3-dimensional convex polyhedra as polynomial images of R3. Internat. J. Math. 25 (2014), no. 7, 1450071 (18 pages). [FU4] J.F. Fernando, C. Ueno: A short proof for the open quadrant problem. J. Symbolic Comput. 79 (2017), no. 1, 57–64. [FU5] J.F. Fernando, C. Ueno: On complements of convex polyhedra as polynomial images of Rn. Discrete Comput. Geom. 62 (2019), no. 2, 292–347. [G] J.M. Gamboa: Reelle Algebraische Geometrie, June, 10th − 16th (1990), Oberwolfach. [Ha] D. Handelman: Representing polynomials by positive linear functions on convex (compact) polyhedra, Pacific J. Math. 132 (1988), no. 1, 35–62. [H] M.W. Hirsch: Differential topology. Corrected reprint of the 1976 original. Graduate Texts in Mathematics, 33. Springer-Verlag, New York (1994). [KPS] K. Kubjas, P.A. Parrilo, B. Sturmfels: How to flatten a soccer ball. Homological and computational methods in commutative algebra, 141–162, Springer INdAM Ser., 20, Springer, Cham (2017). [SSS] R. Sanyal, F. Sottile, B. Sturmfels: Orbitopes. Mathematika 57 (2011), no. 2, 275–314. [S] G. Stengle: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann. 207 (1974), 87–97. [U1] C. Ueno: A note on boundaries of open polynomial images of R2. Rev. Mat. Iberoam. 24 (2008), no. 3, 981–988. [U2] C. Ueno: On convex polygons and their complements as images of regular and polynomial maps of R2. J. Pure Appl. Algebra 216 (2012), no. 11, 2436–2448.