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Such numbers exist. Here is a way to construct one of size close to $n$, for $n\ge 2$: Let $a_1 = n + \frac{1}{2}$. Inductively, for each $k > 1$ let $a_k = (\lfloor a_{k-1}^k\rfloor+\frac{1}{2})^{1/ …

The polynomial $c_0 + c_1x + \cdots + c_{n-2}x^{n-2} + x^n(x-y_1)(x-y_2)\cdots(x-y_n)$ with all $y_i > 0$ can only have $2n$ positive real zeros if its $n-2$th derivative has $n+2$ positive real zer …

The question as stated is false, be we can salvage it by answering the following question: what's the biggest possible ratio of the sum of the inner tetrahedron's edges to the sum of the outer tetrahe …

More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any a …

For the first question, the answer is yes, since the category of sheaves of $A$-modules is a Grothendieck Abelian Category. The least obvious condition one has to check is that this category has a gen …

Consider a local parametrization $O(t), A(t), ...$, with $O = O(0), A = A(0), ...$. Salmon is essentially checking that $\frac{d}{dt}(O(t)A(t),O(t)B(t);O(t)C(t),O(t)D(t)) = 0$ at $t = 0$, where $(OA,O …

For the first amoeba you mentioned, I think your equations should be $e^{-kx}\pm e^{−ky}=\pm e^{-k}$, not $e^{-kx}\pm e^{−ky}=\pm e^{0}$. For the main question, I think you should be using an equatio …

We can prove this with a cross ratio chase, but first we need an easy lemma. Lemma: If points $A,B,C,D$ are on one line and $E,F,G,H$ are on another line, then $(A,B;C,D) = (E,F;G,H)$ if and only if …