Research Article
Year 2020, Volume 28, Issue 28, 98 - 109, 14.07.2020

### References

• J.-P. Borel, Suites de longueur minimale associees a un ensemble normal donne, Israel J. Math., 64 (1988), 229-250.
• J.-P. Borel, Polynomes a coeffcients positifs multiples d'un polynome donne, in Cinquante ans de polynomes (Paris, 1988), Lecture Notes in Math., Springer, Berlin, 1415 (1990), 97-115.
• H. Brunotte, A remark on roots of polynomials with positive coeffcients, Manuscripta Math., 129 (2009), 523-524.
• H. Brunotte, On real polynomials without nonnegative roots, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 26 (2010), 31-34.
• H. Brunotte, On expanding real polynomials with a given factor, Publ. Math. Debrecen, 83 (2013), 161-178.
• H. Brunotte, Polynomials with nonnegative coeffcients and a given factor, Period. Math. Hungar., 66 (2013), 61-72.
• H. Brunotte, On some classes of polynomials with nonnegative coeffcients and a given factor, Period. Math. Hungar., 67 (2013), 15-32.
• H. Brunotte, On canonical representatives of small integers, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 30 (2014), 1-15.
• I. Dancs, Remarks on a paper of P. Turan, Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 7 (1964), 133-141.
• V. De Angelis, Asymptotic expansions and positivity of coeffcients for large powers of analytic functions, Int. J. Math. Math. Sci., (2003), 1003-1025.
• H. G. Diamond and M. Essen, Functions with non-negative convolutions, J. Math. Anal. Appl., 63 (1978), 463-489.
• A. Dubickas, On roots of polynomials with positive coeffcients, Manuscripta Math., 123 (2007), 353-356.
• P. Kirschenhofer and J. M. Thuswaldner, Shift radix systems-a survey, in Numeration and substitution 2012, RIMS Kokyuroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, (2014), 1-59.
• G. Kuba, Several types of algebraic numbers on the unit circle, Arch. Math. (Basel), 85 (2005), 70-78.
• V. Laohakosol and S. Tadee, Algebraic numbers satisfying polynomials with positive rational coeffcients, J. Numbers, (2014), 1-8.
• E. Meissner, Uber positive Darstellungen von Polynomen, Math. Ann., 70 (1911), 223-235.
• T. S. Motzkin and E. G. Straus, Divisors of polynomials and power series with positive coeffcients, Paci c J. Math., 29 (1969), 641-652.
• V. Ponomarenko, Arithmetic of semigroup semirings, Ukrainian Math. J., 67 (2015), 243-266. Reprint of Ukrain. Mat. Zh. 67(2) (2015), 213-229.
• I. SageMath, CoCalc Collaborative Computation Online, 2017. https://cocalc.com/.
• C. Tan and W.-K. To, Characterization of polynomials whose large powers have all positive coeffcients, Proc. Amer. Math. Soc., 146 (2018), 589-600.
• T. Zaimi, On roots of polynomials with positive coeffcients, Publ. Inst. Math. (Beograd) (N.S.), 89(103) (2011), 89-93.

### INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS

Year 2020, Volume 28, Issue 28, 98 - 109, 14.07.2020

### Abstract

For a given real polynomial $f$ without nonnegative roots we study monic integer polynomials $g$ such that the product $g f$ has positive (nonnegative, respectively) coefficients. We show that monic integer polynomials~$g$ with these properties can effectively be computed, and we give lower and upper bounds for their degrees. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

### References

• J.-P. Borel, Suites de longueur minimale associees a un ensemble normal donne, Israel J. Math., 64 (1988), 229-250.
• J.-P. Borel, Polynomes a coeffcients positifs multiples d'un polynome donne, in Cinquante ans de polynomes (Paris, 1988), Lecture Notes in Math., Springer, Berlin, 1415 (1990), 97-115.
• H. Brunotte, A remark on roots of polynomials with positive coeffcients, Manuscripta Math., 129 (2009), 523-524.
• H. Brunotte, On real polynomials without nonnegative roots, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 26 (2010), 31-34.
• H. Brunotte, On expanding real polynomials with a given factor, Publ. Math. Debrecen, 83 (2013), 161-178.
• H. Brunotte, Polynomials with nonnegative coeffcients and a given factor, Period. Math. Hungar., 66 (2013), 61-72.
• H. Brunotte, On some classes of polynomials with nonnegative coeffcients and a given factor, Period. Math. Hungar., 67 (2013), 15-32.
• H. Brunotte, On canonical representatives of small integers, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 30 (2014), 1-15.
• I. Dancs, Remarks on a paper of P. Turan, Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 7 (1964), 133-141.
• V. De Angelis, Asymptotic expansions and positivity of coeffcients for large powers of analytic functions, Int. J. Math. Math. Sci., (2003), 1003-1025.
• H. G. Diamond and M. Essen, Functions with non-negative convolutions, J. Math. Anal. Appl., 63 (1978), 463-489.
• A. Dubickas, On roots of polynomials with positive coeffcients, Manuscripta Math., 123 (2007), 353-356.
• P. Kirschenhofer and J. M. Thuswaldner, Shift radix systems-a survey, in Numeration and substitution 2012, RIMS Kokyuroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, (2014), 1-59.
• G. Kuba, Several types of algebraic numbers on the unit circle, Arch. Math. (Basel), 85 (2005), 70-78.
• V. Laohakosol and S. Tadee, Algebraic numbers satisfying polynomials with positive rational coeffcients, J. Numbers, (2014), 1-8.
• E. Meissner, Uber positive Darstellungen von Polynomen, Math. Ann., 70 (1911), 223-235.
• T. S. Motzkin and E. G. Straus, Divisors of polynomials and power series with positive coeffcients, Paci c J. Math., 29 (1969), 641-652.
• V. Ponomarenko, Arithmetic of semigroup semirings, Ukrainian Math. J., 67 (2015), 243-266. Reprint of Ukrain. Mat. Zh. 67(2) (2015), 213-229.
• I. SageMath, CoCalc Collaborative Computation Online, 2017. https://cocalc.com/.
• C. Tan and W.-K. To, Characterization of polynomials whose large powers have all positive coeffcients, Proc. Amer. Math. Soc., 146 (2018), 589-600.
• T. Zaimi, On roots of polynomials with positive coeffcients, Publ. Inst. Math. (Beograd) (N.S.), 89(103) (2011), 89-93.

### Details

Primary Language English Mathematics Articles Horst BRUNOTTE This is me (Primary Author) Haus-Endt-Stra{\ss}e Germany July 14, 2020 Year 2020, Volume 28, Issue 28

### Cite

 Bibtex @research article { ieja768184, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2020}, volume = {28}, number = {28}, pages = {98 - 109}, doi = {10.24330/ieja.768184}, title = {INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS}, key = {cite}, author = {Brunotte, Horst} } APA Brunotte, H. (2020). INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS . International Electronic Journal of Algebra , 28 (28) , 98-109 . DOI: 10.24330/ieja.768184 MLA Brunotte, H. "INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS" . International Electronic Journal of Algebra 28 (2020 ): 98-109 Chicago Brunotte, H. "INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS". International Electronic Journal of Algebra 28 (2020 ): 98-109 RIS TY - JOUR T1 - INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS AU - HorstBrunotte Y1 - 2020 PY - 2020 N1 - doi: 10.24330/ieja.768184 DO - 10.24330/ieja.768184 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 98 EP - 109 VL - 28 IS - 28 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.768184 UR - https://doi.org/10.24330/ieja.768184 Y2 - 2022 ER - EndNote %0 International Electronic Journal of Algebra INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS %A Horst Brunotte %T INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS %D 2020 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 28 %N 28 %R doi: 10.24330/ieja.768184 %U 10.24330/ieja.768184 ISNAD Brunotte, Horst . "INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS". International Electronic Journal of Algebra 28 / 28 (July 2020): 98-109 . https://doi.org/10.24330/ieja.768184 AMA Brunotte H. INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS. IEJA. 2020; 28(28): 98-109. Vancouver Brunotte H. INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS. International Electronic Journal of Algebra. 2020; 28(28): 98-109. IEEE H. Brunotte , "INTEGER MULTIPLIERS OF REAL POLYNOMIALS WITHOUT NONNEGATIVE ROOTS", International Electronic Journal of Algebra, vol. 28, no. 28, pp. 98-109, Jul. 2020, doi:10.24330/ieja.768184