Infinite series representation for root of polynomials?Given a polynomial p ( x ) = a n x n + ⋯ + a 1 x + a 0 , can every root of the polynomial be represented as ∑ k = 0 ∞ b k with the b k 's being a function of a 0 , … , a n using only elementary operations of arithmetic and taking roots?

Question

Infinite series representation for root of polynomials?Given a polynomial   p  (  x  )  =      a    n        x    n    +  ⋯  +      a    1    x  +      a    0  , can every root of the polynomial be represented as       ∑          k      =      0        ∞        b    k   with the       b    k  &#039;s being a function of       a    0    ,  …  ,      a    n   using only elementary operations of arithmetic and taking roots?

kjakesHB · Accepted Answer

Explanation:I think this is true at least formally if you allow the       b    i   to have coefficients in             Q        ¯  . This is because, if K is an algebraically closed field of characteristic 0, then the field of Puiseux series with coefficients in K is also algebraically closed, and by iterating this construction for each coefficient       a    i   I think we get the desired result abstractly, although I am not sure what one can say about actual (as opposed to formal) convergence.

Jase Sawyer · Accepted Answer

Step 1The difficult part is to get a good a priori estimate   Ω  ⊂            C       of the set S of roots. Starting with any       z    0    ∈  Ω, e.g., with rational coordinates, Newton&#039;s rule      z          n      +      1        :=      z    n    −                    p        (                  z          n                )                              p          ′                (                  z          n                )                (  n  ≥  0  )     ,i.e.,      b    0    =      z    0    ,        b          n      +      1        :=  −                    p        (                  z          n                )                              p          ′                (                  z          n                )                (  n  ≥  0  )  ,Step 2provides a series       ∑          k      ≥      0            b    k   converging to a point   ζ  ∈  S where the       b    k   depend rationally on the coefficients of p (and the chosen point       z    0  ).

Infinite series representation for root of polynomials? Given a polynomial p(x)=a_nx^n+ cdots +a_1x+a_0, can every root of the polynomial be represented as sum_{k=0}^{infty}b_k with the b_k's being a function of a_0,…,a_n using only elementary operations of arithmetic and taking roots?

Answered question

Answer & Explanation

New Questions in Algebra II