Inverse of General Quintic by Determinants

MAIN I will work out how to potentially express the series reversion for a general quintic (and possibly the conditions that this fails) using the determinants of matrices. Let us define our quintic polynomial P(x) = a_0 + a_1x + a_2x^2+a_3x^3 + a_4x^4 + a_5x^5 we can treat this is a series and do a series reversion about a₀ giving P^{-1}(x) = {a_1} - {a_1^3} + {a_1^5}+ {a_1^7} + \cdots there is a clear pattern for the denominators and the (x − a₀) term. Can we express this sequence in the form P^{-1}(x) = ^\infty |M_n|}{a_1^{2n+1}} where Mn is an n × n matrix, and | ⋅ | is the determinant. Then we have M₀ as the empty matrix, define the determinant |M₀|=1, then M_1=-[a_2]\\ M_2=- a_1 & a_2\\ 2a_2 & a_3 \\ M_3= - a_1 & a_2 & 0 \\ 0 & a_1 & 5a_2 \\ a_2 & a_3 & a_4 \\ M_4 = - a_1 & a_2 & 2a_3/3 & 0 \\ 0 & 6a_1 & 14a_2 & 18a_3 \\ 0 & 0 & a_1/6 & a_2 \\ a_2 & a_3 & a_4 & a_5 \\ M_5 = - a_1 & a_2 & a_3 & a_4 & 0 \\ 0 & 7a_1 & 14a_2 & 7a_3 & 0 \\ 0 & 0 & a_1 & 3a_2 & a_3 \\ 0& 0 & 0 & a_1 & a_2 \\ a_2 & a_3 & a_4 & a_5 & 0 we could also swap rows and columns being careful about sign, there are likely other (potentially prettier) solutions. If we could work out a general pattern, then we would have a form for the inverse of a general quintic. However, it can be seen that if a₁ → 0, the inverse function becomes ill defined. INVERSE OF LESSER POLYNOMIALS We might quickly want to apply this method to lesser polynomials, for example the linear inverse P_1(x) = a_0 + a_1 x then we have the inverse by series reversion P_^{-1}(x) = {a_1} we can note this is also the first term of the quintic expression above. For the quadratic we have P_2(x) = a_0 + a_1 x + a_2 x^2 and by reversion the sequence P_2^{-1}(x) = {a_1} - {a_1^3} + {a_1^5} - {a_1^7} + \cdots where the coefficients 1, 1, 2, 5, ⋯ are the Catalan numbers, A000108, which gives a form for this expression P_2^{-1}(x) = ^\infty {n!(n+1)!}}{a_1^{2n+1}} = }{2a_2} which clearly resembles the quadratic formula, this however gives little opportunity to yield a determinant structure, it is clear that the a₂n coefficients are arriving from the determinant. Next we may try the cubic equation P_3(x) = a_0 + a_1x + a_2 x^2 + a_3 x^3 with reversion P_3^{-1}(x) = {a_1} - {a_1^3} + {a_1^5} + {a_1^7} + \cdots this is sharing three terms with the quintic equation reversion, it also gives rise to the use of the determinants. The coefficient of a₂n in each term appears to be the Catalan numbers as in P₂−1(x). We can try to derive the series of matricies ₃Mn which describe the cubic pattern, _3M_1=-[a_2]\\ _3M_2=- a_1 & 2a_2\\ a_2 & a_3 \\ _3M_3= - a_1 & a_2 & 0 \\ 0 & a_1 & 5a_2 \\ a_2 & a_3 & 0 which are the same, if not similar. We now note that there seems to always be a coeffient that is the current Catalan number in the matrix, and it is on a₂. We might be better off assiging these coefficients to the lower left corner which is consistently a₂, _3M_1=-[a_2]\\ _3M_2=- a_1 & a_2\\ 2a_2 & a_3 \\ _3M_3= - a_1 & a_2 & 0 \\ 0 & a_1 & a_2 \\ 5a_2 & a_3 & 0 \\ _3M_4= - 3a_1 & a_2 & 2a_3/7 & 0 \\ 0 & a_1 & a_2 & a_3 \\ 0 & 0 & a_1 & a_2\\14a_2 & a_3 & 0 & 0 \\ _3M_5= - 7a_1 & -a_2 & -37a_3/126 & 0 & 0\\0 & 3a_1 & a_2 & 2a_3/7 & 0 \\ 0& 0 & a_1 & a_2 & a_3 \\0 & 0 & 0 & a_1 & a_2\\42a_2 & a_3 & 0 & 0 & 0 GENERAL SERIES We could also attempt this from the infinite limit, and then set the corresponding coefficients to 0 afterwards. P(x) = a_0 + a_1 x + a_2 x^2 + \cdots = ^\infty a_kx^k then we can describe the inverse term by P^{-1}(x) = ^\infty {na_1^n}\left((-1)^{s+t+u+\cdots}{s!t!u!\cdots}\left({a_1}\right)^s\left({a_1}\right)^t\cdots\right)(x-a_0)^n where the sum is over all combinations such that, s + 2t + 3u + ⋯ = n − 1 (all positive integers). [Morse Feschbach 1953] For example, when n = 1, the only combination is for s = t = u = ⋯ = 0. When n = 2, then then only combination is s = 1, for n = 3 we may have either, s = 2, t = 0 or s = 0, t = 1, giving two terms etc. The goal at this stage would be to develop a regular family of n × n matrices that can express these rules through their determinants. We may find that the matrices are made of the sum or product of matrices expressing each term.