## STNB2020(34th edition)

### The level of pairs of polynomials

#### Presenters

Alberto Fernandez Boix

#### Abstract

Given a prime number $p$ and a polynomial $f$ with coefficients in $\mathbb{F}_p$ , it is known
that there exists a differential operator δ raising $1/f$ to its $p$th power; attached
to this differential operator there is a numerical invariant, the so–called level, that
provides some interesting information about the hypersurface defined by $f$. For
instance, when $f$ is a cubic homogeneous polynomial defining an elliptic curve
$E$, the level of δ is one if and only if $E$ is ordinary, and two if and only if $E$ is
supersingular [BDSV15]; more generally, when $f$ is a homogeneous polynomial of
degree $2g + 1$ defining a hyperelliptic curve $H$ of genus $g$, the level of δ can still
distinguish whether $H$ is either ordinary or supersingular (but not superspecial)
[BCBFY18].

The purpose of this talk is, on the one hand, discussing a relation between the
level of δ and the notion of stratification for certain non–linear differential equations
recently introduced by van der Put and Top [vdPT15]; on the other hand, extending
the notion of level to that of a pair of polynomials. We compute this level in certain
special cases; finally, we present examples of polynomials g and f such that there
is no differential operator raising $g/f$ to its $p$th power.

The content of this talk is based on joint work with Marc Paul Noordman and
Jaap Top [BNT].

${\it References}$

[BCBFY18] I. Blanco-Chacón, A. F. Boix, S. Fordham, and E. S. Yilmaz. Differential operators
and hyperelliptic curves over finite fields. Finite Fields Appl., 51:351–370, 2018. 1

[BDSV15] A. F. Boix, A. De Stefani, and D. Vanzo. An algorithm for constructing certain
differential operators in positive characteristic. Matematiche (Catania), 70(1):239–
271, 2015. 1

[BNT]
A. F. Boix, M. P. Noordman, and J. Top. The level of pairs of polynomials. Available
at https://arxiv.org/pdf/1903.11311.pdf. 1

[vdPT15]
M. van der Put and J. Top. Stratified order one differential equations in positive
characteristic. J. Symbolic Comput., 68(part 2):308–315, 2015. 1

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