<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\magnification=1200 \centerline{\bf Rational functions with
Grassmann coefficients and} \centerline {\bf invariants of surfaces
in superspace}
                $$ $$
\centerline{\bf Adam Haunch}
\bigskip
  We consider rational functions with coefficients in the Grassmann algebra.
 Such   functions naturally arise in connection with  surfaces
 in superspace, i.e., the space which possesses both commuting and anticommuting coordinates.


   We consider hypersurfaces   in Euclidean superspace of dimension $m|2n$ and
   introduce   {\it shape function} of a surface  as
   $R(z,S)=$ ${\rm Ber}\, (1+zS)$ where  $S$ is the shape operator at a given point
   and
   $z$ is a formal parameter. Here   ${\rm Ber}$ is the analogue of determinant in supercase, which is a rational function
   of matrix entries rather than a polynomial.
  The coefficients of the expansion of the shape function over $z$ are differential-geometric invariants of a surface.
     The shape function should be a ratio of polynomials of degrees $m-1$ and $2n$
  respectively. However, a straightforward calculation  can lead to polynomials of unexpectedly higher degrees
  containing common factors. Elimination of these factors can be a
  problem because  of the presence of zero divisors.
   We develop a mechanism of such an elimination. We study  properties of the shape function.
   In particular we discuss analogues of mean and Gaussian curvatures in supercase and
   suggest a generalisation of the  tube formula.
\bye
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