Fields

A field is a mathematical structure consisting of a set of elements and two binary functions called addition and multiplication, both of which are associative. Addition is commutative and there exists an identity element. Also there exist an additive inverse for every element. Multiplication is not necessarily commutative but there does exist a multiplicative identity. There exist muliplicative inverves for all elements except the additive identity.

Let F be a field with S the set of its elements and its two operation denoted by + and juxtaposition for addition and multiplication. The additive and multiplicative identities are denoted as 0 and1, respectively . The additive inverse of an element x is denoted as −x. The multiplicative inverse of x is denoted as x⁻¹.

A polynomial over F is of the form

P(C, k) = c_nkⁿ + c_n-1k^n-1 + … + c₁k + c₀1

where the coefficients c_j and the base of the polynomical k belong to S and n is a nonnegative integer.

The largest power n in a polynomial is called its degree and the smallest power m with a nonzero coefficient will be called the minimum power of the polynomial, The difference (n−m) will be called the degree span of the polynomial.

The Coefficient Sum
of a Polynomial

Let σ(P(C, k)) be the sum of the coefficients of a polynomial. For polynomials of finite degree span the coefficient sum is well defined and finite. For a polynomial of infinite span the coefficient sum may or may not be defined. For examples

σ(1−2k+4k²−8k³+…+(−2k)^j+…) is not defined
σ(1+2k+4k²+8k³+…+(2k)^j+…) is infinite
σ(1−k/2+k²/4+k³/8+…+(k/2)^j+…) is equal to 2
σ(1−k+k²−k³+…+(−k)^j+…) is not defined
σ(1+k+k²+k³+…+k^j+…) is infinite.

The above series are equal to, respectively, 1/(1+2k), 1/(1−2k), 1/(1+½k), 1/(1+k), and 1/(1−k). These functions evaluated at k=1 are equal to, respectively, (1/3), (−1), 2, (1/2), and +∞.

Note that if σ(P(C, k)) is defined and and finite then

σ(P(C, k)) = P(C, 1)

That is to say, the coefficient sum of a polynomial is equal to its value when evaluated for the multiplicative identity. When σ(P(C, k)) is +∞ it may or may not be equal to P(C, 1).

It is important to note that generally the coefficient sum is defined for the polynomial rather than for the value of the polynomial at a particular value of the base. For example, if the polynomial is 2k²+k over the field of real numbers the coefficient sum is 3=2+1. But for k=2 the polynomial has the value 10 and 10 is a polynomial in powers of ten and its coefficient sum is 1.

It is also important to note that the representation of a real number as a decimal is not unique. For example, the real number 1 can be represented both by 1.000… and 0.999….

The Coefficient Sum of a
Function of a Polynomial

What is sought is a theorem that says something to the effect that the coefficient sum of a function of a polynomial is the value of that function evaluated with the base of the polynomial set equal to the multiplicative identity. The problem, as noted above, is that the coefficient sum might not only be infinite, it may be undefined.

Because of the anomalies illustrated in the above examples the functional form for the proposed theorem should be limited to polynomial functions. Thus

Theorem 1: If Q is a polynomial function of the polynomial N=P(C, k), and hence is ultimately a polynomial function in k the base of N then the coefficient sum σ of that polynomial in k, if it is defined, is given by

σ(Q) = σ(Σ d_jN^j)
= Σ d_jσ(N^j)
= Σ d_j(σ(N))^j)
Σ d_j(P(C, 1)^j)

Binary Functions

Let N=P(C, k) and M=P(D, k) be two polynomials in the base k. Then

σ(N+M) = σ(N) + σ(M)
and
σ(N·M) = σ(N)·σ(M)

Proofs:

Trivially

N+M=P(C+D, k)
and hence
σ(N+M)=σ(P(C, k)) + σ(P(D, k))
= σ(N) + σ(M)

For N·M = P(C, k)·P(D, k) the sequence of coefficient is the same as would occur is C and D were converted into sums and these sums multplied. The conversions of C and D into sums are the same as P(C, 1) and P(D, 1), respectively, and hence their product is σ(N)σ(M)

Polynomials over Two Bases

Such polynomials are of the form

P(E, k, q) = Σ Σ e_ijkⁱq^j

where E is the matrix of the coefficients e_ij.

Again let N=P(C, k) and M=P(D, k) be two polynomials in the base k. Let Q be a polynomial function of N and M; i.e., Q=P(E, N, M). But Q is ultimately a polynomial in k, the base of N and M. Then

Theorem2: σ(Q) = σ(P(E, P(C, k), P(D, k))) = P(E, P(C, 1), P(D, 1))

Illustration

Let N=k+1, M=k² and Q=N²M. Then Q evaluates to k⁴+2k³+k² and hence σ(Q)=4. But σ(N)=2 and σ(M)=1 and thus P(E, 2, 1)=2²1=4.