Abstract
Let \(\mathcal{H}_{d,m}\) denote the real homogeneous polynomials in d variables of degree m. In the final chapter of the book we deal with the truncated moment problem for \(\mathcal{H}_{d,2n}\) on \(\mathbb{R}^{d}\), on the unit sphere S d−1, and on S + d−1. Since S + d−1 is a realization of the projective space \(\mathbb{P}^{d-1}(\mathbb{R})\), the latter is in fact the truncated projective moment problem. The existence problem in this case was considered in Sect. 17.2.
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Bibliography
Bakonyi, M. and H.J. Woerdeman: Matrix Completions, Moments, and Sums of Hermitean Squares, Princeton Univ. Press, Princeton, NJ, 2011.
Bix, R.: Conics and Cubics, Springer-Verlag, New York, 1998.
Blekherman, G.: Nonnegative polynomials and sums of squares, J. Amer. Math. Soc. 25(2012), 617–635.
Choi, M.D. and T.Y. Lam: Extremal positive semidefinite forms, Math. Ann. 231(1977), 1–18.
Choi, M.D., T.Y. Lam and B. Reznick: Real zeros of positive semidefinite forms, Math. Z. 171(1987), 559–580.
di Dio, Ph. and K. Schmüdgen: On the truncated multidimensional moment problem: Carathéodory numbers, ArXiv:1703.01494, to appear in J. Math. Anal. Appl.
Eisenbud, D., M. Green and J. Harris: Cayley–Bacharach theorems and conjectures, Bull. Amer. Math. Soc. 33(1996), 295–324.
Ellison, W.J.: A “Waring problem” for homogeneous forms, Cambridge Phil. Soc. 65(1969), 663–672.
Ellison, W.J.: Waring’s problem, Amer. Math. Monthly 78(1971), 10–36.
Fulton, W.: Algebraic curves: an introduction to algebraic geometry, Addison-Wesley, 1998.
Hilbert, G.: Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32(1888), 342–350.
Hilbert, G.: Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Zahl n-ter Potenzen, Math. Ann. 7(1909), 281–300.
Natanson, M.B.: Additive number theory, Springer-Verlag, New York, 1996.
Petrovskii, I.G. and O.A. Oleinik: On the topology of real algebraic surfaces, Isvestiya Akad. Nauk SSSR Ser. Mat. 13(1949), 389–402; Amer. Math. Translation 1992, No. 70.
Ren, Q., J. Richter-Gebert and B. Sturmfels: Cayley–Bacharach formulas, Amer. Math. Monthly 122(2015), 845–854.
Reznick, B.: Sums of Even Powers of Linear Forms, Memoirs Amer. Math. Soc. 9, Providence, R.I., 1982.
Reznick, B.: Homogeneous polynomial solutions to constant PDE’s, Adv. Math. 117(1996), 179–192.
Reznick, B.: On Hilbert’s construction of positive polynomials, arXiv:0707.2156.
C. Riener and M. Schweighofer: Optimization approaches to quadrature: new characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions, arXiv:1607.08404.
Robinson, R.M.: Some definite polynomials which are not sums of squares of real polynomials, in: Selected Questions of Algebra and Logic, Acad. Sci. USSR, 1973, 264–282.
Schmüdgen, K.: A positive polynomial which is not a sum of squares. A positive, but not strongly positive functional, Math. Nachr. 88(1979), 385–390.
Schmüdgen, K.: On the multi-dimensional truncated moment problem: maximal masses, Methods Funct. Anal. Topology 21(2015), 266–281.
Semple, J.G. and L. Roth: Introduction to Algebraic Geometry, Clarendon Press, Oxford, 1949.
Vegter, G.: The apolar bilinear form in geometric modeling, Math. Comp. 69(1999), 691–720.
Walker, R.J.: Algebraic curves, Springer-Verlag, New York, 1978.
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Schmüdgen, K. (2017). The Truncated Moment Problem for Homogeneous Polynomials. In: The Moment Problem. Graduate Texts in Mathematics, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-319-64546-9_19
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DOI: https://doi.org/10.1007/978-3-319-64546-9_19
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