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A007878
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Number of terms in discriminant of generic polynomial of degree n.
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10
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1, 2, 5, 16, 59, 246, 1103, 5247, 26059, 133881, 706799, 3815311, 20979619, 117178725, 663316190, 3798697446, 21976689397
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OFFSET
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1,2
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COMMENTS
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Here "generic" means that the coefficients are algebraically independent symbols. - Robert Israel, Oct 02 2015
At one point it was suggested that this is the same sequence as A039744, but this is wrong. Dean Hickerson, Dec 16 2006, comments as follows: (Start)
The claim that A039744 equals the number of monomials in the discriminant is false. The first counterexample is n=4: There are 18 such partitions, but the discriminant has no terms corresponding to the partitions 3+2+2+2+2+1 and 2+2+2+2+2+2, so the number of monomials in the discriminant is only 16.
Columns near the left or right have very few nonzero elements and this adds some restrictions to the partitions.
For example, from column 2 of the matrix, we see that the partition must have at least one term equal to n or n-1. From the last column, it must have at least one term equal to 0 or 1. Maybe the complete list of such conditions is enough; I don't know.
Even if we could figure out exactly which partitions correspond to monomials that occur in the expansion, I can't rule out the possibility that the coefficients of some such monomial could cancel out, further reducing the number of nonzero monomials in the discriminant. (End)
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LINKS
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EXAMPLE
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Discriminant of a_0 + a_1 x + ... + a_n x^n is 1/a_n times the determinant of a particular matrix; for n=4 that matrix is
[ a_4 a_3 a_2 a_1 a_0 0 0 ]
[ 0 a_4 a_3 a_2 a_1 a_0 0 ]
[ 0 0 a_4 a_3 a_2 a_1 a_0 ]
[ 4a_4 3a_3 2a_2 1a_1 0 0 0 ]
[ 0 4a_4 3a_3 2a_2 1a_1 0 0 ]
[ 0 0 4a_4 3a_3 2a_2 1a_1 0 ]
[ 0 0 0 4a_4 3a_3 2a_2 1a_1 ]
It is easy to see that there are no monomials in the expansion of this involving either a_4 * a_3 * a_2^4 * a_1 or a_4 * a_2^6.
The discriminant of the cubic K3*x^3 + K2*x^2 + K1*x + K0 is -27*K3^2*K0^2 + 18*K3*K2*K1*K0 - 4*K2^3*K0 - 4*K3*K1^3 + K2^2*K1^2 which contains 5 monomials. - Bill Daly (bill.daly(AT)tradition.co.uk)
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MAPLE
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A007878 := proc(n) local x, a, ii; nops(discrim(sum(a[ ii ]*x^ii, ii=0..n), x)) end;
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MATHEMATICA
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Clear[f, g]; g[0] = f[0]; g[n_Integer?Positive] := g[n] = g[n - 1] + f[n] x^n; myFun[n_Integer?Positive] := Length@Resultant[g[n], D[g[n], x], x, Method -> "BezoutMatrix"]; Table[myFun[n], {n, 1, 8}] (* Artur Jasinski, improved by Jean-Marc Gulliet (jeanmarc.gulliet(AT)gmail.com) *)
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PROG
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(Magma) function Disc(n) F := FunctionField(Rationals(), n); R<x> := PolynomialRing(F); f := x^n + &+[ (F.i)*x^(n-i) : i in [ 1..n ] ]; return Discriminant(f); end function; [ #Monomials(Numerator(Disc(n))) : n in [ 1..7 ] ] // Victor S. Miller, Dec 16 2006
(Sage)
A = InfinitePolynomialRing(QQ, 'a')
a = A.gen()
for N in range(1, 7):
x = polygen(A, 'x')
P = sum(a[i] * x^i for i in range(N + 1))
M = P.sylvester_matrix(diff(P, x), x)
print(M.determinant().number_of_terms())
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CROSSREFS
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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reiner(AT)math.umn.edu
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EXTENSIONS
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a(9) from Lyle Ramshaw (ramshaw(AT)pa.dec.com)
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STATUS
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approved
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