# Questions tagged [ideals]

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### Extracting implications in polynomial constraint system from Groebner basis

Given a Groebner basis for a system of polynomial constraints over $\mathbb{Q}$, are there any known methods for extracting the low degree factorable polynomials in the ideal generated by that basis? ...
1 vote
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### Gorenstein property from initial ideal

My question is: If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...
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### If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k[t]$?

Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$. Let $f,g,h \in k[x,y]$, $g \neq h$, satisfy the following two conditions: (1) $(f,g)$ is a maximal ideal of ...
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### $u,v \in k[x,y]$ that satisfy: For every automorphic image $w \in k[x,y]$, there exist $a,b,c \in k$ such that $(u-a,v-b,w-c)$ is a maximal ideal

Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$, and let $u,v \in k[x,y]-k$. Denote $(u,v)$ the ideal generated by $u$ and $v$. Assume that the following ...
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### Proof that the left ideal I of prime norm in maximal order can be written as I = ON + Oα

It seems there is a well-known fact that if $O$ is a maximal order in quaternion algebra $B$ and $I$ is a left $O-$ideal such that $nrd(I) = N$ is prime, then $I = ON + Oα$ with $gcd(N^2, nrd(α)) = N$....