User santiago - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:56:17Z http://mathoverflow.net/feeds/user/13782 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121010/is-there-any-o-minimal-expansion-of-the-real-field-with-functions-of-growth-highe Is there any o-minimal expansion of the real field with functions of growth higher than exponential? Santiago 2013-02-06T19:39:57Z 2013-02-07T11:33:42Z <p>Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,&lt;)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there $f(x)$ is ultimately greater than $\exp^N(x)$. That is there is we can find $r\in \mathbb{R}$ such that for all $x>k$ we have $\exp^N(x) &lt; f(x)$ where $\exp^N(x)$ stands for $\exp(\exp(\cdots \exp(x)\cdots))$ $N$ times. </p> <p>Is it possible to find an o-minimal expansion (in the model theoretic sense) of $\bar{\mathbb{R}}$ where an $f$ as above is definable?</p> http://mathoverflow.net/questions/73659/properties-from-tropical-geometry-that-do-not-imply-their-algebraic-counterpart Properties from Tropical Geometry that do not imply their algebraic counterpart. Santiago 2011-08-25T13:34:58Z 2011-08-25T15:43:18Z <p>One of the motivations to study tropical geometry is that there are some hard Algebraic Questions that can be answered by proving them in the Tropical World. For example one can show that tropical Bezout's Theorem implies the Algebraic Bezout. </p> <p>What properties are there known that are true (or might be) in tropical geometry that don't imply that their algebraic version is true?</p> http://mathoverflow.net/questions/58984/tropical-properties-from-algebraic-geometry Tropical Properties From Algebraic Geometry Santiago 2011-03-20T17:15:27Z 2011-05-11T02:41:31Z <p>What properties of tropical geometry (Starting from a valued Field) can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series $\mathbb{C}((t))$ what can be said about $\Gamma ^n$ where $\Gamma$ is the valuation group.</p> <p>That is what pairs of properties $(X,X')$ such that $X$ is valid on an algebraic variety then $X'$ is valid on the tropicalization, and $X$ is a first order property in the language $L={ 0,1,+,*,U,|}$ where $U$ stands for the valuation ring, and $x|y \leftrightarrow \exists z U(z) _\wedge xz=y$ </p> <p>A silly example would be the following: One can prove that when the support (the set of exponents corresponding to non zero coefficients) of a polynomial $f$ in two variables is equal to {$(i,j)\in \mathbb{N}^2 | i+j \leq d$} for some $d\in \mathbb{Z}^+$ and the support of another polynomial $g$ is {$(i,j)\in \mathbb{N}^2 | i+j \leq c$} for another $c\in \mathbb{Z}^+$ then the tropical curves generated by those polynomials intersects in exactly $cd$ points or in infinitely many(Bézout). </p> http://mathoverflow.net/questions/64559/bases-of-ideals-with-no-monomials Bases of Ideals With no Monomials Santiago 2011-05-11T02:30:14Z 2011-05-11T02:30:14Z <p>Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in $J$. Is there any characterization on a finite set of generators (probably a reduced Gröbner base) $G$ of $J$?</p> <p>To rephrase my question, Is there a way to know when an ideal in $K[\underline{x}]$ has no monomials by just looking at a set of finitely many generators?</p> http://mathoverflow.net/questions/58979/completeness-of-algebraically-closed-valued-fieldsacvf-theory Completeness of Algebraically Closed Valued Fields(ACVF) Theory Santiago 2011-03-20T15:34:25Z 2011-03-20T20:21:20Z <p>One can prove Elimination of Quantifiers of ACVF finding an extension of any partial embedding of a model $K$ into a $|K|^+$ Saturated one using the language $\mathcal{L} = ( 0,1,+,*, U, \mid )$. In this Language $U$ is the unary predicate standing for the Valuation Ring of the model, and $\mid$ is a binary relation such that $x\mid y \leftrightarrow \exists z\in U \ x*z=y$. How do you prove the completeness of this theory in that language?</p>