Structures of precision losses in computing approximate Gröbner bases (Q1946967)

The author presents a new theory for the structure of precision losses in computing approximate Gröbner bases. Contrary to the existing methods such as interval method (Shirayanagi and Sweedler) and double-float arithmetic (Traverso and Zanoni), his method is based on the fact that the precision losses of the coefficients in one approximate polynomial are dependent. Using a concept of PL-space, the author proves that any PL-space of a polynomial has a finite weak basis containing the minimal forms of precision losses.