• Title, Summary, Keyword: differential polynomial

### EXISTENCE OF POLYNOMIAL INTEGRATING FACTORS

• Stallworth, Daniel T.;Roush, Fred W.
• Kyungpook Mathematical Journal
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• v.28 no.2
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• pp.185-196
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• 1988
• We study existence of polynomial integrating factors and solutions F(x, y)=c of first order nonlinear differential equations. We characterize the homogeneous case, and give algorithms for finding existence of and a basis for polynomial solutions of linear difference and differential equations and rational solutions or linear differential equations with polynomial coefficients. We relate singularities to nature of the solution. Solution of differential equations in closed form to some degree might be called more an art than a science: The investigator can try a number of methods and for a number of classes of equations these methods always work. In particular integrating factors are tricky to find. An analogous but simpler situation exists for integrating inclosed form, where for instance there exists a criterion for when an exponential integral can be found in closed form. In this paper we make a beginning in several directions on these problems, for 2 variable ordinary differential equations. The case of exact differentials reduces immediately to quadrature. The next step is perhaps that of a polynomial integrating factor, our main study. Here we are able to provide necessary conditions based on related homogeneous equations which probably suffice to decide existence in most cases. As part of our investigations we provide complete algorithms for existence of and finding a basis for polynomial solutions of linear differential and difference equations with polynomial coefficients, also rational solutions for such differential equations. Our goal would be a method for decidability of whether any differential equation Mdx+Mdy=0 with polynomial M, N has algebraic solutions(or an undecidability proof). We reduce the question of all solutions algebraic to singularities but have not yet found a definite procedure to find their type. We begin with general results on the set of all polynomial solutions and integrating factors. Consider a differential equation Mdx+Ndy where M, N are nonreal polynomials in x, y with no common factor. When does there exist an integrating factor u which is (i) polynomial (ii) rational? In case (i) the solution F(x, y)=c will be a polynomial. We assume all functions here are complex analytic polynomial in some open set.

### MODULES OVER THE $\phi$- DIFFERENTIAL POLYNOMIAL RINGS

• Sohn, Mun-Gu;Rim, Seog-Hoon
• Bulletin of the Korean Mathematical Society
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• v.22 no.1
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• pp.1-5
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• 1985
• The differential polynomial ring A[X, D] has been studied by many authors J. Cozzens, C. Faith, R.E. Johnson and D. Mathis and others. The main purpose of the present paper is to study some properties of .phi.-differential polynomial ring A[X, D, .phi.] and modules over the .phi.-differential polynomial ring X[X, D, .phi.].

### ON ENTIRE SOLUTIONS OF NONLINEAR DIFFERENCE-DIFFERENTIAL EQUATIONS

• Wang, Songmin;Li, Sheng
• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1471-1479
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• 2013
• In this paper, we study the non-existence of finite order entire solutions of nonlinear differential-difference of the form $$f^n+Q(z,f)=h$$, where $n{\geq}2$ is an integer, $Q(z,f)$ is a differential-difference polynomial in $f$ with polynomial coefficients, and $h$ is a meromorphic function of order ${\leq}1$.

### BRÜCK CONJECTURE AND A LINEAR DIFFERENTIAL POLYNOMIAL

• Lahiri, Indrajit;Pal, Bipul
• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.799-808
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• 2018
• In the paper we consider the uniqueness of a meromorphic function and a linear differential polynomial when they share a small function.

### RESULTS ON UNIQUENESS OF DIFFERENTIAL MONOMIAL AND DIFFERENTIAL POLYNOMIAL

• WAGHAMORE, HARINA P.;VIJAYLAXMI, S.B.
• Journal of applied mathematics & informatics
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• v.37 no.3_4
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• pp.281-294
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• 2019
• In this paper, we study the results concerning uniqueness of meromorphic functions sharing a small function and present one theorem which extend and improves previous results.

### SOME RESULTS ON A DIFFERENTIAL POLYNOMIAL RING OVER A REDUCED RING

• Han, Jun-Cheol;Kim, Hong-Kee;Lee, Yang
• East Asian mathematical journal
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• v.16 no.1
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• pp.89-96
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• 2000
• In this paper, a differential polynomial ring $R[x;\delta]$ of ring R with a derivation $\delta$ are investigated as follows: For a reduced ring R, a ring R is Baer(resp. quasi-Baer, p.q.-Baer, p.p.-ring) if and only if the differential polynomial ring $R[x;\delta]$ is Baer(resp. quasi-Baer, p.q.-Baer, p.p.-ring).

### SOME GROWTH ASPECTS OF SPECIAL TYPE OF DIFFERENTIAL POLYNOMIAL GENERATED BY ENTIRE AND MEROMORPHIC FUNCTIONS ON THE BASIS OF THEIR RELATIVE (p, q)-TH ORDERS

• Biswas, Tanmay
• Korean Journal of Mathematics
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• v.27 no.4
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• pp.899-927
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• 2019
• In this paper we establish some results depending on the comparative growth properties of composite entire and meromorphic functions using relative (p, q)-th order and relative (p, q)-th lower order where p, q are any two positive integers and that of a special type of differential polynomial generated by one of the factors.

### DIFFERENTIAL EQUATIONS RELATED TO FAMILY A

• Li, Ping;Meng, Yong
• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.247-260
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• 2011
• Let h be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if $f^m$ is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type $$f^n+P(f)=be^{sz}+Q(e^z)$$, where P(f) is a differential polynomial in f of degree at most n-1, and Q($e^z$) is a polynomial in $e^z$ of degree k $\leqslant$ max {n-1, s(n-1)/n} with small functions of $e^z$ as its coefficients.

### SLOWLY CHANGING FUNCTION ORIENTED GROWTH MEASUREMENT OF DIFFERENTIAL POLYNOMIAL AND DIFFERENTIAL MONOMIAL

• Biswas, Tanmay
• Korean Journal of Mathematics
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• v.27 no.1
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• pp.17-51
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• 2019
• In the paper we establish some new results depending on the comparative growth properties of composite entire and meromorphic functions using relative $_pL^*$-order, relative $_pL^*$-lower order and differential monomials, differential polynomials generated by one of the factors.

### ON COMMUTING ORDINARY DIFFERENTIAL OPERATORS WITH POLYNOMIAL COEFFICIENTS CORRESPONDING TO SPECTRAL CURVES OF GENUS TWO

• Davletshina, Valentina N.;Mironov, Andrey E.
• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1669-1675
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• 2017
• The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.