15
votes
Accepted
Construction of a symmetric polynomial in the roots that acts like the discriminant
In characteristic not equal to $2$, the discriminant is optimal. In characteristic $2$, the polynomial $\prod_{i<j} (x_i+x_j)$ works and has degree $\binom{n}{2}$.
Proof: Let $f(x_1, x_2, \ldots, ...
- 156k
11
votes
Accepted
What is the essence of the constant factor in the standard definitions of the discriminant?
As Robert said, if you want everything to work in $\mathbb Z[f_0,\ldots,f_m]$, you need that factor. I'll also mention that your polynomial indexing is messed up, you probably meant the sum to go from ...
- 47.4k
10
votes
What is the essence of the constant factor in the standard definitions of the discriminant?
The factor $f_0^{2m-2}$ makes the discriminant a polynomial in the coefficients $f_0, \ldots, f_m$.
- 54.2k
9
votes
Accepted
What is the subgroup of $\mathrm{SL}(n,\mathbb{C})$ which preserves the discriminant?
At the request of the OP, I post my comment as an answer. View $\mathcal{P}_n$ as the space of homogeneous polynomials of degree $n$ in 2 variables. Let $\Delta _p\subset \mathcal{P}_n$ be the locus ...
- 38k
7
votes
Accepted
Sum of derivative of polynomial over its simple roots
I don't know if these sums have a name, but you can compute them using resultants.
Let $P$ be a degree-$d$ polynomial over $\mathbb{C}$ with not-necessarily-simple roots $\alpha_1, \ldots, \alpha_d \...
- 336
7
votes
Accepted
Why the sign in the definition of the discriminant?
The reason is that the formula on the right side should be considered more fundamental, not the formula on the left, when seeking a symmetric expression in the roots. Don't use a product of anything &...
- 50.6k
7
votes
Accepted
Discriminants of some $q$-analogs of $(1+x)^n$
This is true.
We have
\begin{align*}
p_n (q^{-1}, 1-r, x) &= \sum_{j=0}^n q^{ (r-1) \binom{j}{2}} \binom{n}{j}_{q^{-1}} x^j \\
&= \sum_{j=0}^n q^{ (r-1) \binom{j}{2}} q^{-j (n-j)} \binom{n}...
- 148k
7
votes
Accepted
Definition of a Discriminant in Three Variables
Given $n$ homogeneous polynomials $F_i(x_1,\ldots,x_n)$ in $n$ variables with respective degrees $d_i$, there is a unique polynomial ${\rm Res}(F_1,\ldots,F_n)$ in the coefficients of the $F_i$ called ...
6
votes
Accepted
Is the discriminant of a free (as a module) $R$-algebra always congruent to a square modulo 4?
I see this question is still being viewed. To avoid misleading anyone, let me put a placeholder answer here: The question has been answered positively (with a beautiful proof) by John Voight, Asher ...
- 33.8k
5
votes
Accepted
What would be a standard reference for the formula of the discriminant of $f(t^d)$?
You say "presumably not difficult to prove." Did you try? It seems like a pretty easy exercise, using
$$ \Delta(F(t)) = \prod_{F(a)=0} F'(a). $$
Taking $F(t)=f(t^d)$, the roots of $F$ are the $d$'th ...
- 47.4k
4
votes
Accepted
Show that $\sum_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}$ is separable
Let me elaborate on Noam D. Elkies' comment. If we denote $n=(p-1)/2$, the discriminant of this polynomial $g(x)$ is non-zero modulo $p$ if and only if the discriminant of Legendre's polynomial $f(x)=...
- 109k
4
votes
Is the set of integers represented by a quadratic form of non-fundamental discriminant a subset of the rep. set of a form of fundamental discriminant?
This is true, of course. The book you want is Binary Quadratic Forms by D. A. Buell. He talks about how, given some discriminant $\Delta$ and class number(primitive forms) $h(\Delta),$ we can predict $...
- 25.7k
4
votes
Accepted
Discriminant of a radical extension of a quadratic number field
Write $D_n$ for the absolute discriminant $\left|\mathrm{disc}(L_n|\mathbb{Q})\right|$ of the field of interest and $d_n$ for its degree. Then the root discriminant is bracketed in the interval $$5^{...
- 2,391
4
votes
Accepted
Upper bound for discriminant of Galois closure
One approach is to prove the corresponding fact on the Artin conductor side. Let $G$ be the Galois group ok $\tilde{K}/\mathbb Q$ and $H$ the Galois group of $\tilde{K}/K$, so that $|G|= \tilde{N}$ ...
- 148k
4
votes
What is the essence of the constant factor in the standard definitions of the discriminant?
Perhaps you will find the following helpful, which expands a bit on Robert Israel's answer: Write $f(x)=f_0x^m+\ldots+f_m=f_0\cdot(x-\alpha_1)\cdots(x-\alpha_m).$ Multiplying out the RHS we get $f(x)...
- 868
3
votes
A mysterious expression from a discriminant
It might be illuminating to write the discriminant as the resultant of your polynomial with its derivative. If I've done the algebra correctly (but please check), the discriminant that you're ...
- 47.4k
3
votes
What would be a standard reference for the formula of the discriminant of $f(t^d)$?
I don't know of a standard reference, but there is a note by John Cullinan at Bard College where he computes the discriminant of a composition of two polynomials, so a nontrivially more general result ...
- 96.4k
2
votes
What is the essence of the constant factor in the standard definitions of the discriminant?
A trivial observation, which was not pointed out so far: the non-normalized discriminant is 'better' in that it comes 'closer' to being a 'homomorphism' $P[x]\rightarrow P$, though it 'still' is not ...
- 6,051
2
votes
Discriminants of Clifford algebras
It was proved in Theorem 3.7 of "Discriminant Formulas and Applications" by Chan, Young and Zhang.
- 71
1
vote
Why the sign in the definition of the discriminant?
The definition of the resultant $\text{res}(f,g)$ of two (monic) polynomials $f,g\in {\mathbb Z}[x]$ as the determinant of the corresponding Sylvester matrix, readily implies
that if $\alpha_1,\ldots,\...
- 1,561
1
vote
Show that $\sum_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}$ is separable
This is not an answer, but rather a reformulation. Write $\equiv_p$ for congruence mod $p$. Here $p$ is odd.
Start with $\binom{\frac{p-1}2}i\equiv_p\binom{2i}i4^{-i} \mod p$. If we drop $x^{\frac{p-...
- 43.2k
1
vote
Accepted
A generalization of the discriminant of a polynomial
As to question (2): Suppose that $\Delta_f(y)$ is separable. Then $\Delta_f(y)$ is irreducible if and only if the Galois group of $f$ acts transitively on the set of differences $\alpha_i-\alpha_j$, $...
- 22.5k
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