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Yes, every real number $u \in [0,1]$ can be written as $u = x^2 y$ where $x,y \in C$ are in the Cantor set $C$. In particular, every real number in $[0,1]$ is a product of three Cantor set elements. T …

I believe this would be a dual arrangement of a star arrangement. A star arrangement is a union of subspaces defined as follows. Let $H_1,\dotsc,H_d$ be a collection of hyperplanes and fix an integer …

Yes, it is true. Every polynomial does have a decomposition like you ask for, with restrictions. To explain why this is the case, first suppose that $p$ is homogeneous of degree $d$, and we seek a dec …

The $p=2$ dimensional case is an exercise in Rogawski's calculus textbook. It is exercise 47 on page 885, section 15.1 (Integration in Several Variables) in the 2008 Early Transcendentals edition.

That is the Noether-Lefschetz theorem. Searching online should find plenty of results in web pages and lecture notes. If you want a published source, how about: Mark Green, A new proof of the explicit …

Here are a few articles that might be relevant: Narasimhan, The irreducibility of ladder determinantal varieties, from 1986. It takes about 20 pages to prove the result (that the minors are a standard …

Consider the symmetric group action permuting the variables. The Vandermonde determinant $V$ is antisymmetric, meaning it spans an alternating representation—it's invariant under permutations, up to m …

AMS has started a list, organized by mathematical field, at https://www.ams.org/profession/online-talks.

Two books by David Bressoud: Bressoud, David M. Proofs and Confirmations. The story of the alternating sign matrix conjecture. MAA Spectrum. 1999 "My intention in this book is not just to descr …

Your condition (1) means: if $\hat{u}$, $\hat{v}$, and $\hat{w}$ are linearly dependent, then so are $\widehat{\varphi(u)}$, $\widehat{\varphi(v)}$, and $\widehat{\varphi(w)}$. So $\varphi$ preserves …

Since each $F_i(u_1,\dotsc,u_{n_1},v_1,\dotsc,v_{n_2})=0$, then $F_i(su_1,\dotsc,su_{n_1},tv_1,\dotsc,tv_{n_2}) = s^{d_1}t^{d_2} F_i(u_1,\dotsc,u_{n_1},v_1,\dotsc,v_{n_2})=s^{d_1}t^{d_2}0 = 0$. So ce …

All values of $(i,\lambda_i)$ appear. The ones that aren't outside corners are redundant.

This paper might possibly be relevant: http://www.ams.org/mathscinet-getitem?mr=2922602 It's not quite exactly what you are asking for. Instead of Gorenstein algebras of low rank, they are consideri …