# All Questions

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### Multiplicity of a point in positive characteristic

Let $C$ be a plane curve over a field of characteristic $p>0$ and let $x\in C$ be a point. How is the multiplicity of $C$ at $x$ calculated (in char = 0 one uses partial derivatives) ?
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### existence of a special conformal mapping

Sorry I don't know how to give an appropriate title. In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...
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### Why is $\psi(\lambda)$ a complex linear polynomial?

Let $\phi(\lambda)$ be an entire complex function with bounded modulus. Now we know that $\phi(\lambda)$ is constant. If, however, we know that $$\phi(\lambda) = \exp[\psi(\lambda)]$$ for some entire ...
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### How can I turn this series notation to a formula? [on hold]

I have the summation SIGMA(0 to h) of ((h+1)(2^h)). I need a way to convert this in to a formula f(h) = x where x is the right answer. I remember doing arithmetic and geometric series' in maths in ...
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### algebraic leaves of foliation on a product of two curves

Let $S=E\times C$ be a product of two curves, where $E$ is an elliptic curve and $C$ is a curve of genus at least two. Consider a foliation on $S$ generated by a global holomorphic 1-form ...
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### Tensor product of fields over integers

Inspired by this question we ask; Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$?: 1.A field $K$ with the property that ...
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### What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...
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### Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...
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### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...
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### Two to the power of a triangular number: bijections

The numbers $2^{n(n+1)/2}$ come up in various enumerative contexts. In addition to the trivial example (bit-strings of length $n(n+1)/2$) and the old example of domino tilings of Aztec diamonds ...
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### Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$. In Macaulay's words, the ...
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### Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...
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### Reference request: Systems of linear PDES with constant coefficients

I am looking for a reference for the following statement: Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs \begin{align} P_i(\partial / \partial x_1, \dots, ...
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### Are there any good computer programs for drawing (algebraic) curves?

I realise that I lack some intuition into how a curve (or surface, or whatever) looks geometrically, from just looking at the equation. Thus, I sometimes resort to some computer program (such as ...
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### “You can't push a rope” [on hold]

"You can't push a rope" is a wisdom saying that some engineering teachers pass along to their students. Since I'm not an engineer, I can only guess at what they mean, but it sounds to me like code ...
Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by ...
### Homotopy groups of spheres in a $(\infty, 1)$-topos
Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces). You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, ...