invariant polynomial initial
DESCRIPTION
Invariant polynomials.TRANSCRIPT
Initial Formula
I wanted to do this in LaTex instead of just typing it into the email, foreasiness in reading(and also to brush up my Tex skills, which I haven’t usedfor a few weeks now).
We will begin with the complex function, G(z).
G(z) = zn = (x + iy)n
Expanding this further, we see:
G(z) =
(n
0
)xny0i0 +
(n
1
)xn−1y1i1 +
(n
2
)xn−2y2i2 + . . . +
(n
n
)x0ynin
Then, for the polynomial R(x, y), we have that:
R(x, y) =
(n
0
)xn−
(n
2
)xn−2y2+
(n
4
)xn−4y4−
(n
6
)xn−6y6+. . .
(n
2j
)xn−2jy2j
With j = bn2c ∈ N
A note here, I’m not quite sure how to notate the end of the polynomial.My aim is to show that the polynomial Isn’t endless, and that the exponenton x is greater than or equal to 0, as per the definition of a polynomial.The issue is that whether the last term of the polynomial has x1 or x0 isdependent on whether n is even or odd. I tried to use the notation withthe floor function defining a value for j, and hopefully that’s obvious, butis there a better way for me to notate this that I’m missing?
The polynomial I(x, y) is similar, it’s the leftover from the original com-plex function:
I(x, y) =
(n
1
)xn−1y1−
(n
3
)xn−3y3+
(n
5
)xn−5y5−
(n
7
)xn−7y7+. . .
(n
2k − 1
)xn−2k+1y2k+1
With k = dn2e
The same problem exists here, where I don’t quite know how to notate thefinal term. I’m also fairly certain that, if not for both, then at least forI(x, y), the exponents on x and y are incorrect by 1, again simply becauseI couldn’t think of a good way to notate it(maybe because of the time ofnight).
Are these polynomials correct? Are these more invariants? Also, arethese a finite number of these invariant polynomials for A? Should I alsobe trying to prove that I have all of them or something similar to that, oris it too early?
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