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%I #72 Nov 09 2023 08:54:22
%S 1,2,5,16,59,246,1103,5247,26059,133881,706799,3815311,20979619,
%T 117178725
%N Number of terms in discriminant of generic polynomial of degree n.
%C Here "generic" means that the coefficients are algebraically independent symbols. - _Robert Israel_, Oct 02 2015
%C 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)
%C 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.
%C Columns near the left or right have very few nonzero elements and this adds some restrictions to the partitions.
%C 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.
%C 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)
%H Kinji Kimura, <a href="http://syskiso.fuee.u-fukui.ac.jp/~kkimur/susemi.html">programs written in Maple, Magma, etc.</a>
%e 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
%e [ a_4...a_3...a_2...a_1...a_0...0.....0... ]
%e [ 0.....a_4...a_3...a_2...a_1...a_0...0... ]
%e [ 0.....0.....a_4...a_3...a_2...a_1...a_0. ]
%e [ 4a_4..3a_3..2a_2..1a_1..0.....0.....0... ]
%e [ 0.....4a_4..3a_3..2a_2..1a_1..0.....0... ]
%e [ 0.....0.....4a_4..3a_3..2a_2..1a_1..0... ]
%e [ 0.....0.....0.....4a_4..3a_3..2a_2..1a_1 ]
%e 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.
%e 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)
%p A007878 := proc(n) local x,a,ii; nops(discrim(sum(a[ ii ]*x^ii, ii=0..n), x)) end;
%t 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) *)
%o (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
%o (Sage)
%o A = InfinitePolynomialRing(QQ, 'a')
%o a = A.gen()
%o for N in range(1, 7):
%o x = polygen(A, 'x')
%o P = sum(a[i] * x^i for i in range(N + 1))
%o M = P.sylvester_matrix(diff(P, x), x)
%o print(M.determinant().number_of_terms())
%o # _Georg Muntingh_, Jan 17 2014
%K nonn,nice,hard,more
%O 1,2
%A reiner(AT)math.umn.edu
%E a(9) from Lyle Ramshaw (ramshaw(AT)pa.dec.com)
%E Entry revised by _N. J. A. Sloane_, Dec 16 2006
%E a(10) from _Artur Jasinski_, Apr 02 2008
%E a(11) from _Georg Muntingh_, Jan 17 2014
%E a(12) from _Georg Muntingh_, Mar 10 2014
%E a(13)-a(14) from _Seiichi Manyama_, Nov 08 2023
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