D’Angelo, John P.; Kos, Šimon; Riehl, Emily A sharp bound for the degree of proper monomial mappings between balls. (English) Zbl 1052.26016 J. Geom. Anal. 13, No. 4, 581-593 (2003). In this interesting paper the authors prove the following theorem. Suppose that \(p\) is a polynomial in two real variables \((x,y)\) with real coefficients, such that \(p(x,y)=1\), if \(x+y=1\). Let \(N\) be the number of distinct monomials in \(p\), and let \(d\) be the degree of \(p\). Then \(d\leq 2N-3\) and this result is sharp. As a corollary, they show that if \(f\) is a proper holomorphic monomial mapping from the unit ball in \(\mathbb C^2\) to the unit ball in \(\mathbb C^N\), then the degree of \(f\) does not exceed \(2N-3\), and this result is sharp. Reviewer: Wiesław Pleśniak (Kraków) Cited in 3 ReviewsCited in 18 Documents MSC: 26C05 Real polynomials: analytic properties, etc. 32B99 Local analytic geometry 11B83 Special sequences and polynomials 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables Keywords:proper holomorphic mappings; Lucas polynomial; monomials PDFBibTeX XMLCite \textit{J. P. D'Angelo} et al., J. Geom. Anal. 13, No. 4, 581--593 (2003; Zbl 1052.26016) Full Text: DOI Online Encyclopedia of Integer Sequences: Odd degrees for which (up to swapping of variables) there exists a unique polynomial p(x,y), such that p(x,y)=1 when x+y=1, with positive coefficients and such that the number of terms is minimal (equal to (d+3)/2). There always exists a group invariant polynomial (see any of the references), but for many degrees, other such extremal polynomials exist. Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y) = 1 whenever x + y = 1; a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree, i.e., of degree 2n-3. Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y)=1 whenever x + y = 1. a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree minus 1, i.e., of degree 2n-4. Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y)=1 whenever x + y = 1. a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree minus two, i.e., of degree 2n-5. References: [1] Alexander, H., Proper holomorphic mappings in C_n, Indiana Univ. Math. J., 26, 137-146 (1977) · Zbl 0391.32015 [2] D’Angelo, J. P., Several Complex Variables and the Geometry of Real Hypersurfaces (1993), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 0854.32001 [3] D’Angelo, J. P., Proper polynomial mappings between balls, Duke Math. J., 57, 211-219 (1988) · Zbl 0657.32012 [4] D’Angelo, J. P., Invariant holomorphic mappings, J. Geom. Anal., 6, 2, 163-179 (1996) · Zbl 0901.32017 [5] D’Angelo, J. P.; Lichtblau, D., Spherical space forms, CR mappings, and proper maps between balls, J. Geom. Anal., 2, 5, 391-415 (1992) · Zbl 0782.32016 [6] Faran, J. J., Maps from the two-ball to the three-ball, Inv. Math., 68, 441-475 (1982) · Zbl 0519.32016 [7] Kos, Šimon, Two applications of the quasiclassical method to superfluids, Thesis (2001), Urbana: Dept. of Physics, University of Illinois, Urbana [8] Koshy, T., Fibonacci and Lucas Numbers (2001), NY: John Wiley & Sons, NY · Zbl 0984.11010 [9] Pinchuk, S., On the holomorphic continuation of holomorphic mappings, Math. USSR-Sb., 27, 375-392 (1975) · Zbl 0366.32010 [10] Setya-Budhi, W. Proper holomorphic mappings in several complex variables, (thesis), University of Illinois, (1993). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.