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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.
9
votes
0
answers
351
views
Which polynomials in the minors of a matrix are invariant under conjugation?
These $\text{GL}(\bigwedge^k \mathbb{R}^n)$
-invariant polynomials are classified. … Perhaps this is not a coincidence-maybe all $\text{GL}(n)$-invariant polynomials of the $k$-minors are "factored" through the determinant in some way. …
9
votes
2
answers
455
views
Can we recover all $k$-minors of a square matrix from some of them?
This is a cross-post.
Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" invertible $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of $A$). …
3
votes
1
answer
133
views
Is the smallest root of this quartic always the closest point on the Hyperbola? [closed]
Let $a>b>0$.
Suppose we want to minimize
$$
f(x)=(x-a)^2+(1/x-b)^2,
$$
over $x>0$.
Equating $f'(x)=0$ leads to the quartic equation
$$
g(x)=x^4-ax^3+bx-1=0. \tag{1}
$$
Question:
Is the smallest positi …