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Results tagged with polynomials
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user 38468
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.
33
votes
Accepted
The sum of squared logarithms conjecture
Is there anything wrong with the following argument?
First of all, by scaling all $x_i$ and all $y_i$ by a positive constant, we may safely assume that $\prod_i x_i = \prod_i y_i =1$.
The result wou …
- 5,186
5
votes
Accepted
Is there a critical point of a polynomial $f$ within every disc having as diameter the line ...
No. Take $f(z)=z^n-1$ to get an easy counterexample.
- 5,186
4
votes
Accepted
Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of genera...
Containment in $I_i$ is a codimension one condition, and there is a
dimension $(l+1)(l+2)/2$ space of degree $l$ polynomials, so such $F_1$ exists. … Consider a polynomial $F_2$ which does not have to be homogeneous, but rather is a sum of polynomials of degrees $\geq l$ which is contained in all $I_i$ (and presumably is generic with respect to this …
- 5,186
4
votes
Dimension of a homogeneous polynomial system
I don't have a complete solution, but the following may be helpful.
Change variables by $z_i = \sum_j y_j \xi^{ij}$ where $\xi$ is $m$-th primitive root of $1$.
Then the first line equations (I am us …
- 5,186
3
votes
When are Ehrhart functions of compact convex sets polynomials?
I believe that the strong form of the conjecture is false. In lieu of a simple counterexample, let me point you towards a centrally symmetric 10-gon $\hat P$ in arXiv:0801.2812, Figure 6. It is a bit …
- 5,186
3
votes
Accepted
Polynomial vector field tangent to a given analytic simple closed curve
I see this as very unlikely. A polynomial vector field would have a slope function that is a rational function of two variables with a finite number of coefficients.
As a consequence, if you take th …
- 5,186