List of proposed problems

These are the possible problems to work on jointly during the 2008/2009 season. See also the List of proposed topics.

Feel free to expand, comment, ask…

### Problem 1: Real Resolution

Consider the example of the cone. It has a singular point at the origin and a resolution is given by the cylinder. One can define a proper map, bianalytic outside of the origin resp. a circle, between the cone and the cylinder, thus providing a "real resolution" (both varieties live in $\mathbb{R}^3$). A challenge will be to find different examples of such a real resolution and a general statement telling when this is possible.

• This problem seems interesting. Is there a plan to split the work, or to work jointly in some way? — David

### Problem 2: Is Luengo's Lemma true?

It seems that the following lemma is used in some paper(s) but lacking a complete proof. Our work would be to try to understand what is missing and find a counterexample or fix the proof.

Luengo's Lemma: Let $F=F(x_1,..., x_n,y) \in k[[x_1,...,x_n,y]]$ be regular in y, $deg_y F = d$, and assume that $disc_y(F) = x_1^{m_1}...x_n^{m_n} .u(x), u(0)\neq 0$, $char k=0$. Moreover assume $F$ is Tschirnhaus. Then the projection of the Newton polytope of $F$ from $(0,...,0,d)$ to the $\{y=0\}$-plane is an orthant.

First we'll have to ask the experts if a proof or counteexample are known. If neither is the fact, well, then we have a "problem".

### Problem 3: Mathematics in robotics?

We can try to solve some problems related to robotics (could be other field of applications as well ?) where a lot of mathematics are involved. I could for example describe an old or actual problem and not mention the solution. We could work on it and see what we can come up with. Looking at these kind or problems from a purely mathematical point of view using advanced mathematical tools (which engineers are often not aware of) can lead to complete different formulations which can help solve the problem or give a deeper understanding of the phenomena. Material could come from the conferences ARK and Workshop on Parallel Mechanisms that I will attend in 2 weeks. These problems do not normally require much knowledge about robotics, since they can normally can be expressed purely algebraically and geometrically.

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