17
votes
Accepted
Find a special integer coefficients polynomial which takes small absolute value on [0,4]
No. Integer coefficients are a red herring; the point is that, if $f(x) = f_n x^n + \cdots + f_0$ is a real polynomial with $f_n \neq 0$, then the $L_{\infty}$ norm of $f(x)$ on an interval of the ...
- 149k
4
votes
Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
There are likely more-elementary approaches, but here is an argument using some basic function-field arithmetic. Let's work a bit more generally than in the original question, considering a base field ...
- 41
3
votes
Accepted
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
Discriminant is the resultant
of $f$ and $f'$. In your case, it is simply an integer.
If this integer is $\neq 0$ then $f$ has no multiple root
(and vice versa).
- 87.2k
2
votes
Signed factors of harmonic polynomials
It seems that Dennis Serre has already addressed the question. I just want to point out that the result is a special case of a theorem of Brelot and Choquet.
Suppose that $h$ is a (not necessarily ...
- 181
2
votes
Accepted
Formulas for partial composed product
Thanks everyone for the comments! I will try to summarize what's going on a bit, and post it as an answer, as I was not really aware of how symmetric squares and exterior squares work, so I will ...
- 1,111
1
vote
Accepted
Slicing bivariate exponential generating functions on x and y
Assuming $D(0) = 0$, we can restate the problem as $F(x, y) = A(x)^y$ for $A(x) = e^{D(x)}$.
It means that we can find $G_n(k)$ for any number $k$ as
$$
G_n(k) = \left[\frac{x^n}{n!}\right] A(x)^k.
$$
...
- 1,111
1
vote
Difference between Chebyshev first and second degree iterative methods
Late to the party by 7 years but here goes.
This is easiest to analyze in the case where $A$ is symmetric and positive-definite.
What you're describing is the difference between using the steepest ...
- 628
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