Discriminants of the depressed polynomial of degree $n$

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Polynomials

$n$

$\operatorname{disc}(f_n)$

-4*a0

-4*a1^3 - 27*a0^2

-4*a1^2*a2^3 + 16*a0*a2^4 - 27*a1^4 + 144*a0*a1^2*a2 - 128*a0^2*a2^2 + 256*a0^3

-4*a1^2*a2^2*a3^3 + 16*a0*a2^3*a3^3 + 16*a1^3*a3^4 - 72*a0*a1*a2*a3^4 + 108*a0^2*a3^5 - 27*a1^2*a2^4 + 108*a0*a2^5 + 144*a1^3*a2^2*a3 - 630*a0*a1*a2^3*a3 - 128*a1^4*a3^2 + 560*a0*a1^2*a2*a3^2 + 825*a0^2*a2^2*a3^2 - 900*a0^2*a1*a3^3 + 256*a1^5 - 1600*a0*a1^3*a2 + 2250*a0^2*a1*a2^2 + 2000*a0^2*a1^2*a3 - 3750*a0^3*a2*a3 + 3125*a0^4

-4*a1^2*a2^2*a3^2*a4^3 + 16*a0*a2^3*a3^2*a4^3 + 16*a1^3*a3^3*a4^3 - 72*a0*a1*a2*a3^3*a4^3 + 108*a0^2*a3^4*a4^3 + 16*a1^2*a2^3*a4^4 - 64*a0*a2^4*a4^4 - 72*a1^3*a2*a3*a4^4 + 320*a0*a1*a2^2*a3*a4^4 + 24*a0*a1^2*a3^2*a4^4 - 576*a0^2*a2*a3^2*a4^4 + 108*a1^4*a4^5 - 576*a0*a1^2*a2*a4^5 + 512*a0^2*a2^2*a4^5 + 768*a0^2*a1*a3*a4^5 - 1024*a0^3*a4^6 - 27*a1^2*a2^2*a3^4 + 108*a0*a2^3*a3^4 + 108*a1^3*a3^5 - 486*a0*a1*a2*a3^5 + 729*a0^2*a3^6 + 144*a1^2*a2^3*a3^2*a4 - 576*a0*a2^4*a3^2*a4 - 630*a1^3*a2*a3^3*a4 + 2808*a0*a1*a2^2*a3^3*a4 + 162*a0*a1^2*a3^4*a4 - 4860*a0^2*a2*a3^4*a4 - 128*a1^2*a2^4*a4^2 + 512*a0*a2^5*a4^2 + 560*a1^3*a2^2*a3*a4^2 - 2496*a0*a1*a2^3*a3*a4^2 + 825*a1^4*a3^2*a4^2 - 4536*a0*a1^2*a2*a3^2*a4^2 + 8208*a0^2*a2^2*a3^2*a4^2 + 5832*a0^2*a1*a3^3*a4^2 - 900*a1^4*a2*a4^3 + 4816*a0*a1^2*a2^2*a4^3 - 4352*a0^2*a2^3*a4^3 - 120*a0*a1^3*a3*a4^3 - 5760*a0^2*a1*a2*a3*a4^3 - 8640*a0^3*a3^2*a4^3 - 192*a0^2*a1^2*a4^4 + 9216*a0^3*a2*a4^4 + 256*a1^2*a2^5 - 1024*a0*a2^6 - 1600*a1^3*a2^3*a3 + 6912*a0*a1*a2^4*a3 + 2250*a1^4*a2*a3^2 - 9720*a0*a1^2*a2^2*a3^2 - 8640*a0^2*a2^3*a3^2 - 1350*a0*a1^3*a3^3 + 21384*a0^2*a1*a2*a3^3 - 8748*a0^3*a3^4 + 2000*a1^4*a2^2*a4 - 10560*a0*a1^2*a2^3*a4 + 9216*a0^2*a2^4*a4 - 3750*a1^5*a3*a4 + 19800*a0*a1^3*a2*a3*a4 - 3456*a0^2*a1*a2^2*a3*a4 - 27540*a0^2*a1^2*a3^2*a4 + 3888*a0^3*a2*a3^2*a4 + 1500*a0*a1^4*a4^2 - 6480*a0^2*a1^2*a2*a4^2 - 17280*a0^3*a2^2*a4^2 + 46656*a0^3*a1*a3*a4^2 - 13824*a0^4*a4^3 + 3125*a1^6 - 22500*a0*a1^4*a2 + 43200*a0^2*a1^2*a2^2 - 13824*a0^3*a2^3 + 27000*a0^2*a1^3*a3 - 77760*a0^3*a1*a2*a3 + 34992*a0^4*a3^2 - 32400*a0^3*a1^2*a4 + 62208*a0^4*a2*a4 - 46656*a0^5

Definition

For $n\geq2$, let $f_n(x)=x^n+\sum_{i=0}^{n-2} a_i x^i$ be the monic depressed polynomial with formal coefficients $a_0,\dots,a_{n-2}$ [1]. The entry is $\operatorname{disc}(f_n)$ [2], normalized by Formula (1).

Parameters

$n$

— degree ($n\geq2$)

Formulas

(1)

$\operatorname{disc}(f_n)=(-1)^{n(n-1)/2}\operatorname{Res}(f_n,f_n')$, where the resultant is normalized as in [3] by $\operatorname{Res}(f,g)=\prod_{f(\alpha)=0}g(\alpha)$ for monic $f$.

(2)

Equivalently, if $f_n(x)=\prod_{j=1}^n(x-\alpha_j)$, then $\operatorname{disc}(f_n)=\prod_{1\leq i<j\leq n}(\alpha_i-\alpha_j)^2$.

(3)

$\operatorname{disc}(x^2+a_0)=-4a_0$.

(4)

$\operatorname{disc}(x^3+a_1x+a_0)=-4a_1^3-27a_0^2$.

Comments

(5)

The variable $a_i$ multiplies $x^i$, so $a_0$ is the constant term and the variables in the degree-$n$ entry are $a_0,\dots,a_{n-2}$.

(6)

The depressed-polynomial discriminant is the specialization of the discriminant of the general degree-$n$ polynomial for $f(x)=\sum_{i=0}^n a_i x^i$, with the general polynomial's $a_n$ set to $1$ and $a_{n-1}$ set to $0$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

n = 5
R = PolynomialRing(ZZ, ["a%s" % i for i in range(n - 1)])
a = R.gens()
S = PolynomialRing(R, "x")
x = S.gen()
f = x**n + sum(a[i] * x**i for i in range(n - 1))
sign = -1 if (n * (n - 1) // 2) % 2 else 1
sign * f.resultant(f.derivative())

Links

[1]

Wikipedia: Depressed polynomial

[2]

Wikipedia: Discriminant

[3]

Wikipedia: Resultant

Similar tables

Discriminants of the general polynomial of degree $n$ — gives the same discriminant before specializing the general polynomial's leading coefficient to $1$ and the coefficient of $x^{n-1}$ to $0$

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every degree $2\leq n\leq6$; the degree $7$ entry has 8099 characters and 320 terms, too long to read on a page)

How they were obtained:

The generator computes exact integer coefficients in $\mathbb Z[a_0,\dots,a_{n-2}]$ from the resultant identity in Formula (1), and all arithmetic is exact.

It checks Formula (2) by substituting the elementary symmetric polynomials of roots satisfying $\alpha_1+\cdots+\alpha_n=0$, compares the quadratic and cubic rows from Formula (3) and Formula (4) against the stored values, compares degrees $2$ through $5$ with the specialization of the general-polynomial discriminant, and compares integer specializations with Sage's exact univariate discriminant over $\mathbb Z[x]$.