sorry, I misspoke myself. The situation is more complicated than I thought. 
It is true that the original file contains only true arc elements, no Bezier 
curves. But when you edit it, you see the handles typical of Bezier curves, you 
do not see the handles typical of an arc. (using the Node tool to draw this 
distinction) The reason apparently is that the object has no sodipodi 
attributes, which suggests that perhaps it was not created by Inkscape. In the 
absence of sodipodi attributes, the editing and probably also the rendering 
will be done using Bezier curves.
    So the question is, why is the Bezier approximation so poor. The reason is 
that the arc angle is too high. In the curved portion of the 'p', the outer 
curve section is represented by only 2 Bezier curves and they both have an arc 
angle greater than 90 degrees. The Bezier approximation to an arc very rapidly 
becomes very poor if the arc angle is allowed to be higher than 90 degrees.
    So the next question is, why is file p_new.svg better, after manually 
expanding the image? Because this file contains 8 Bezier curves, not 2, in the 
same curved portion of the 'p' character.
    I suspect, but obviously cannot prove, that the approximation procedure 
that converts from arc to Bezier probably has some numerical cutoff that 
determines how many segments to use and that this cutoff value is in absolute 
units not scaled units, so that when you convert a tiny image (200 times 
smaller than normal) you get fewer Bezier segments than when you convert a 
normal size image, leading to a situation where a very tiny image is not as 
well-formed as a normal size image.

-- 
Bezier curves drawn, where svg file requires circular arcs
https://bugs.launchpad.net/bugs/653315
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