Genus of algebraic curves with unknown degree - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:47:05Z http://mathoverflow.net/feeds/question/75093 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75093/genus-of-algebraic-curves-with-unknown-degree Genus of algebraic curves with unknown degree unknown (yahoo) 2011-09-10T12:23:36Z 2011-09-11T06:46:57Z <p>I am not sure if this is a valid question but posting any way:</p> <p>Say I am over $\mathbb{F}_{p}$ for a prime $p$.</p> <p>I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown form (and hence degree). How many points do I need to know on the curve to estimate the genus of the curve? </p> <p>$A.)$We also have the additional constraint that $f(y)$ has atmost $2k$ non-zero coeffients where $k$ is a constant. Assume that a bound to the degree is known.</p> <p>$B.)$Assume $2k &lt;&lt; y$-degree of the equation.</p> <p>$C.)$Assume the coeffients of the highest half of the terms are $+1$ and the lowest half of terms are $-1$. (Just an artificial example - but this tells that one possibly may be able to get the genus without getting the coefficients. For a concrete realization of the artificial example, look at error correction codes over $3$ alphabets $\{ \pm1, 0 \}$. The errors can be in only $2k$ coordinates and I also know the errors in the top half will be $+1$ and the lower half will have errors with $-1$).</p> <p>How many points do you need? If degree bound is $D$, then would $O(\log^{h(k)}{D})$ points suffice where $h(k)$ is independent of $D$ and of the curve and is fixed for a fixed $k$? </p> http://mathoverflow.net/questions/75093/genus-of-algebraic-curves-with-unknown-degree/75098#75098 Answer by Felipe Voloch for Genus of algebraic curves with unknown degree Felipe Voloch 2011-09-10T13:46:30Z 2011-09-10T15:31:51Z <p>No finite set will do. If $f(y)$ is one such polynomial and $cy^m$ is a monomial occurring in it, then change this monomial to $cy^{m+q-1}$ for some power $q$ of $p$ such that your finite set is a subset of the field of $q$ elements. The two curves will go through the same finite set of points but will have different genera if $q$ is sufficiently large. </p> <p>However, now the OP has decided to change the question.</p>