Joint differential resolvents for pseudopolynomials

The existence of linear differential resolvents for z for any root z of an ordinary polynomial with coefficients in a given ordinary differential field has been established, where α is an indeterminate constant with respect to the derivation of the given field. In this paper we consider several alphas. We will call a finite sum of indeterminate powers of a variable v a pseudopolynomial in v . We will generalize the definition of a differential resolvent of a single polynomial for a single monomial z to the definition of a differential resolvent of several polynomials for a pseudopolynomial in the roots. We will also generalize the definition of a resolvent to have non-consecutive derivatives. We will show that the author’s powersum formula may be used to compute this more general differential resolvent. 1. Standard notation from differential algebra and Introduction All rings will be commutative with unity 1. Let denote the ring of integers. Let + denote the set of positive integers. Let 0 + denote the set of nonnegative integers. Let denote the field of rational numbers. For any finite set S , let S denote its size. The symbol ∀ means “for all”. The symbol ∋ means “such that”. The symbol ≡ means “is defined as” or “is identically equal to”. For any positive integer, m + ∈ , define [ ] { 1 } m k k m + ≡ ∈ ∋ ≤ ≤ and 0 0 [ ] { 0 } m k k m + ≡ ∈ ∋ ≤ ≤ . For any ring , let # denote the proper subset of nonzero elements of . For any ring , let 1 [ ,..., ] L h h denote the ring generated by and the variables 1 { ,..., } L h h , that is, the set of all polynomials over of 1 { ,..., } L h h . For any field F , let 1 ( ,..., ) L h h F denote the field generated by F and the variables 1 { ,..., } L h h , that is, the set of all rational functions over F of the variables 1 { ,..., } L h h . A derivation D on a ring is a map from to which satisfies the Leibniz rule ( ) ( ) D u v u Dv D u v ⋅ = ⋅ + ⋅ . If r∈ , then r D ⋅ is also a derivation on . Hence, 0 D ⋅ , which maps to 0, is a derivation. Henceforth, we will assume that our derivation D is not the identically zero derivation, that is, D is non-trivial. A ring equipped with a non-trivial derivation D is called a differential ring. For any differential ring , let 1 { ,..., } L h h denote the differential ring generated by and the variables 1 { ,..., } L h h , that is, the set of all polynomials over of infinitely many derivatives of the variables 1 { ,..., } L h h . For any differential field F , let 1,..., L h h < > F denote the differential field generated by F and the variables 1 { ,..., } L h h , that is, the set of all rational functions over F of infinitely many derivatives of the variables 1 { ,..., } L h h . Although nothing in Theorems 3.1 and 4.2 require the differential fields have zero characteristic, many other theorems on differential resolvents have been proven only for char(F )=0. Hence, unless we specify otherwise, assume zero characteristic for all differential fields. Example 1.5 will demonstrate a resolvent when char(F )=3. Joseph Louis Lagrange published his famous formula for an inverse of a power series of a single variable in 1770 [5]. In Section 5.1 of [2] Egorychev generalized Lagrange’s formula to a formula for multiple variables. Suppose 1 ( , ,..., ) 0 i n F w z z = are n holomorphic functions of the n variables 1 { ,..., } n z z and variables w . There is no loss of generality if we lump all the free variables together and call them by the single letter name w . Egorychev gave increasingly more explicit formulae for ( , ( )) w w φ Φ where Φ is a given holomorphic function of its arguments, and 1 1 ( ,..., ) ( ) ( ( ),..., ( )) n n z z w w w φ φ φ = = are implicit functions determined by 1 ( , ,..., ) 0 i n F w z z = . When the i F are polynomials in the n variables 1 { ,..., } n z z , then there exists a linear differential resolvent – basically, a finite-order linear differential operator – for Φ whose terms depend upon w . We will coin a new term for this differential operator for Φ -we will call it a joint differential resolvent of the polynomials i F for Φ . For any constant α , transcendental or algebraic over 1,..., n z z < > , an α resolvent of a univariate polynomial ( ) P t with n roots 1 { ,..., } n z z is the special case of a differential resolvent of ( ) P t for the pseudomonomial z Φ = . To be consistent with notation on earlier documents on differential resolvents rather than Egorychev’s notation, we will use the letter y instead of Φ henceforth. This article will generalize the definition of a differential α -resolvent into three directions. Generalization 1 of 3. First, we will generalize the definition of a differential resolvent of a single polynomial to the joint differential resolvent of several polynomials. Example 1.1. Let 1( ) P t be a quadratic polynomial with roots 1 x z e = and 2 ln z x = . In other words, 2 1( ) ( ) ( ln ) ( ln ) ln x x x P t t e t x t e x t e x = − ⋅ − = − + ⋅ + ⋅ . Define a derivation D such that 1 Dx ≡ . Let F denote the smallest differential field generated by the coefficients of 1( ) P t , in other words, ln , ln x x e x e x = < + ⋅ > F . We call F the differential coefficient field of 1( ) P t . By definition, a differential α -resolvent of 1( ) P t is any nonzero finite-order linear differential operator R whose terms lie in α < > F such that 0 x e ⋅ R = and (ln ) 0 x α R = . Note that since ln x e x ⋅ ∈F , then ( ln ) x D e x ⋅ ∈F . So 1 ln x x e x e x ⋅ + ⋅ ∈F . So 1 x e x ⋅ ∈F . Since ln x e x + ∈F , then ( ln ) x D e x + ∈F . So 1 x e x + ∈F . The inclusions 1 x e x ⋅ ∈F and 1 x e x + ∈F imply 2 x ∈F , so 2 x ∈F . So 2 ( ) D x ∈F . So 2x∈F . So x∈F . Note also that α constant implies ( ) α α < >= F F , and { } [ ] α α = for any differential ring . Since 1 0 z α R = and 2 0 z α R = , then 1 2 ( ) 0 az bz α α R + = for any constants a and b . In particular, therefore, 1 2 ( ) ( (ln ) ) 0 x z z e x α α α α R + = R + = . We may exploit this fact to quickly compute a resolvent over the smaller differential subfield ( ) x x = < >⊂ F . In general, when α is not an integer, there does not exist an α -resolvent all of whose terms lie in ( ) α F for some smaller differential subfield F of the polynomial 1( ) P t ’s coefficient differential field F . When α is an integer, there often does exist an α -resolvent over a smaller subfield F , but whose order increases with the absolute magnitude of α . In general, the trade-off of desiring a resolvent all of whose terms lie in , ⊆ ≠ F F F F is a resolvent of higher order. For example, let (ln ) x y e x α α ⋅ ≡ + . For each m + ∈Z , , 1 ( ) (ln ) m m m x k m k k k D y e B x α α α α ⋅ − = = + ⋅ ⋅ ∑ where (1.1) 1 2 3 1 1 , , ( , , 2 ,..., ( 1) ( 1)! ,...( 1) ( 1)! ) j j k k m k m k B B x x x j x k x − − − − − − − = − − − ⋅ − − ⋅ is the Bell polynomial in its arguments, which are negative powers of x . The definition of Bell polynomials is given on page 31 of [4]. The falling factorial or Pochhammer symbol ( )k α is defined as 1