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loismustdie
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#1
Report Thread starter 15 years ago
#1
I'm trying to come up with a conjecture about the value of the determinants of matrices W[n]. Problem being that the determinants are horrible polynomials:

detW[1] := 1
detW[2] := x - 1
detW[3] := x (x + 1) (x - 1)^3
detW[4] := x^4 (x - 1)^6 (x + 1)^2 (x^2 + x + 1)
detW[5] := x^10 (x^2 + 1) (x^2 + x + 1)^2 (x + 1)^4 (x - 1)^10
detW[6] := x^20 (x^4 + x^3 + x^2 + x + 1) (x^2 + 1)^2 (x^2 + x + 1)^3 (x + 1)^6 (x - 1)^15
detW[7] := x^35 (x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)^2 (x^2 + 1)^3 (x^2 + x + 1)^5 (x + 1)^9 (x - 1)^21
detW[8] := x^56 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (x^2 - x + 1)^2 (x^4 + x^3 + x^2 + x + 1)^3 (x^2 + 1)^4 (x^2 + x + 1)^7 (x + 1)^12 (x - 1)^28
detW[9] := x^84 (x^4 + 1) (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)^2 (x^2 - x + 1)^3 (x^4 + x^3 + x^2 + x + 1)^4 (x^2 + 1)^6 (x^2 + x + 1)^9 (x + 1)^16 (x - 1)^36

Nice...

Apparently, the fact that (x-1) (x^2 + x + 1) = x^3 - 1 is a hint (?).

Any ideas at all?
If it helps, W is an nxn matrix whose (i-j)th entry is x^((i-1)(j-1)).
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BCHL85
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#2
Report 15 years ago
#2
Can not understand your work
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RichE
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#3
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#3
I would google van der Monde determinant if I were you - as your W[n] is a special case of that determinant.
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J.F.N
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#4
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The vandermonde matrix has a particularly beautiful determinant. I'm not sure what relation your matrix has to a vandermonde matrix (if any; but I would trust RichE's judgment). Are you sure there is some general conjecture for the determinant of W[n]? I've used expansion by minors and there doesn't seem to be a discernable pattern.
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RichE
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#5
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#5
Maybe if I was to rewrite the determinants as:

1
x-1
(x^2-x)(x^2-1)(x-1)
(x^3-x^2)(x^3-x)(x^3-1)(x^2-x)(x^2-1)(x-1)
...

it might become clearer.
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J.F.N
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#6
Report 15 years ago
#6
(Original post by RichE)
Maybe if I was to rewrite the determinants as:

1
x-1
(x^2-x)(x^2-1)(x-1)
(x^3-x^2)(x^3-x)(x^3-1)(x^2-x)(x^2-1)(x-1)
...

it might become clearer.
Brilliant. What level math are you doing?
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