A007878 Number of terms in discriminant of generic polynomial of degree n.

1, 2, 5, 16, 59, 246, 1103, 5247, 26059, 133881, 706799, 3815311, 20979619, 117178725, 663316190, 3798697446, 21976689397

Offset: 1

Author

reiner(AT)math.umn.edu

Keywords

• nonn
• nice
• hard
• more

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)

Examples

			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)
		

Links

• Kinji Kimura, Computing the general discriminant formula of degree 17, presented at Gröbner Bases, Resultants and Linear Algebra workshop, September 3-6, 2013, Research Institute for Symbolic Computation, Hagenberg, Austria. See abstract and author's website.
• Kinji Kimura, programs written in Maple, Magma, etc.

Programs

Magma

function Disc(n) F := FunctionField(Rationals(),n); R := 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

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) *)

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())
# Georg Muntingh, Jan 17 2014

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

a(13)-a(14) from Seiichi Manyama, Nov 08 2023

a(15)-a(17) from Kimura (2013) added by Andrey Zabolotskiy, Jun 30 2024