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An approach to define the resultant of two entire functions. (English) Zbl 1369.30005
Let \(f_n(z)=a_0+a_1z+\dots+ a_nz^n\), \(a_n\neq 0\), \(g_m(z)=b_0+b_1z+\dots+ b_mz^m\), be polynomials. The classical resultant \(R(f_n, g_m)\) of \(f_n\) and \(g_m\) can be written in the form \(R(f_n, g_m)=a_n^m \prod_{k=1}^{n} g_m(\alpha_k)\) where \(\alpha_1, \dots, \alpha_n\) are zeros of \(f_n\). Using this property the authors introduce the resultant of two entire functions, where one of them has finitely many zeros. Some formulas for the calculation of the resultant in terms of determinants are established.

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D10 Representations of entire functions of one complex variable by series and integrals
Full Text: DOI
[1] Van der Waerden BL, Algebra 1 (1950)
[2] Kurosh AG, Course in higher algebra (1968)
[3] DOI: 10.1007/978-1-4613-0041-0 · doi:10.1007/978-1-4613-0041-0
[4] DOI: 10.1007/978-94-011-5302-7 · doi:10.1007/978-94-011-5302-7
[5] DOI: 10.1080/03081088108817420 · Zbl 0584.12018 · doi:10.1080/03081088108817420
[6] DOI: 10.1007/978-0-8176-4771-1 · doi:10.1007/978-0-8176-4771-1
[7] Gohberg IC, Acta Sci. Math 37 pp 41– (1975)
[8] DOI: 10.1007/BF01896778 · Zbl 0341.15011 · doi:10.1007/BF01896778
[9] DOI: 10.1090/S0002-9939-1978-0493487-8 · doi:10.1090/S0002-9939-1978-0493487-8
[10] Gustafsson B, Commun. Math. Phys 10 pp 265– (2009)
[11] DOI: 10.1007/s11232-010-0044-0 · Zbl 1254.15011 · doi:10.1007/s11232-010-0044-0
[12] Bykov VI, Modeling of the critical phenomena in chemical kinetics (2006)
[13] Tsybenova VI, Non-linear models of chemical kinetics (2011)
[14] Jury EI, Inners and stability of dynamic systems (1974) · Zbl 0307.93025
[15] Titchmarsh EC, The theory of functions (1939)
[16] Gantmacher F, The theory of matrices (1959) · Zbl 0085.01001
[17] Macdonald IG, Oxford Mathematical Monograph, in: Symmetric functions and hall polynomials (1979)
[18] Kytmanov A, Complex. Anal. Oper. Th
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