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A007878 Number of terms in discriminant of generic polynomial of degree n. 11
1, 2, 5, 16, 59, 246, 1103, 5247, 26059, 133881, 706799, 3815311 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..12.

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537.  Zbl 1133.11012.

EXAMPLE

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)

MAPLE

A007878 := proc(n) local x, a, ii; nops(discrim(sum(a[ ii ]*x^ii, ii=0..n), x)) end;

MATHEMATICA

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) *)

PROG

(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)

for N in range(1, 7):

....Lvars = ['x'] + ['a'+str(i) for i in range(N+1)]

....R = PolynomialRing(QQ, Lvars)

....R.inject_variables()

....P = sum([globals()["a"+str(i)]*x^i for i in range(N+1)])

....M = P.sylvester_matrix(diff(P, x), x)

....print len(M.determinant().monomials())

# Georg Muntingh, Jan 17 2014

CROSSREFS

Sequence in context: A185143 A280760 A000753 * A019589 A087949 A028333

Adjacent sequences:  A007875 A007876 A007877 * A007879 A007880 A007881

KEYWORD

nonn,nice,hard,more

AUTHOR

reiner(AT)math.umn.edu

EXTENSIONS

a(9) from Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Entry revised by N. J. A. Sloane, Dec 16 2006

a(10) from Artur Jasinski, Apr 02 2008

a(11) from Georg Muntingh, Jan 17 2014

a(12) from Georg Muntingh, Mar 10 2014

STATUS

approved

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Last modified October 16 14:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)