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Results tagged with polynomials
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user 2083
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
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Is $[JK:(x)][JK:(y,z)]\subseteq JK$ in $k[x,y,z]$?
Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $J,K\subseteq m$ be proper homogeneous ideals in $S$. Is this true that we always have:
$$[JK:(x)][JK:(y,z)]\subseteq JK \ ?$$
Some backgroun …
- 30.6k
7
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Accepted
If a polynomial f is irreducible then (f) is radical, without unique factorization?
Here are some thoughts on why such a proof might be hard to find (and interesting). They are not totally rigorous, but I think they give a different perspective, so might be worth something.
I belie …
- 30.6k
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Criteria for system of parameters in polynomial rings
Recall that the resultant $Res(g_1,\cdots, g_n)$ of $n$ (not necessarily homogenous) polynomials in $n-1$ variables is $0$ if and only if the $g_i$s have a common root. …
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4
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Minimum number of generators of the product of ideals
No. $k[x,y]$ we can find an ideal $I$ with $m$ generators for any $m\geq 5$ but $I^2$ has $9$ generators. See this paper.
Yes, at least in the graded case. There is probably a better way to show th …
- 30.6k
10
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A question about an application of Molien's formula to find the generators and relations of ...
As pointed out by Richard Stanley, there are some subtleties in finding the invariant generators.
The Molien series only tells you the Hilbert series of the ring of invariants up to reduced rational …
- 30.6k
78
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9
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Irreducibility of polynomials in two variables
An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in two variables, Acta Arith. 1997).
Does anyone know of similar results in the same vein? …
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5
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Detecting if a polynomial is a Pfaffian
Now that the question makes sense, let me mention the following result (you can find it in Eisenbud, Exercise 20.17) which hinted at what Francesco wrote.
Let $R = k[[x,y,z]]$ and $f\in m=(x,y,z)$. T …
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Given an integer $N$, find solutions to $X^3 + Y^3 + Z^3 - 3XYZ \equiv 1 \pmod{N}$
Since, $X^3+Y^3+Z^3-3XYZ=\frac{1}{2}(X+Y+Z)((X-Y)^2+(Y-Z)^2+(Z-X)^2)$, taking $X,Y,Z$ close to each other give some non-trivial and cheap solutions.
For instance $(k+1,k,k)$ for $N=3k$, $(k+1,k+1,k)$ …
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Polynomials with rational coefficients
Let $f(x)=x^3-5x/4$. Then for $x\neq y$, $f(x)=f(y)$ iff $x^2+xy+y^2=5/4$ or $(2x+y)^2+3y^2=5$. The last equation clealy have real solutions. But if there are rational solutions, then there are intege …
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Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a...
Take $I=(a^3,b^3)$ and $J=(ac^2-bd^2)$. Then according to Macaulay2, $I\cap J$ has generators in degrees $7,8,9$, for instance $a^3c^6-b^3d^6$. So the answers to Question 3 and 1 are no.
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