Gröbner basis computation is used in solving systems of polynomial equations. The basic operations in computing a Gröbner basis are polynomial coefficient addition (subtraction) and multiplication (division). The polynomial coefficients are typically real numbers represented (approximated) by floating-point numbers with finite precision on computers. The precision can change during calculation. Thus, it is important to be able to track the precision losses during the computation of the coefficients.

Liang, in this paper, presents a method for tracking precision in the computation of Gröbner bases. First, the author introduces the &tgr;-representation, which represents a floating-point number along with its precision. For example, 1.10010010000111010 × 2¹ &tgr;³ represents a floating-point number of precision length of 18 binary bits whose least significant three bits are inaccurate. Thus, &tgr;³ represents the precision loss of three bits in the floating-point number. Liang then discusses the arithmetic operations addition (subtraction) and multiplication (division) of the &tgr; representations. For example, a¹ &tgr;^A₁ × a₂ &tgr;^A₂ = (a₁ a₂) &tgr;^{max (A₁, A₂)} (assumption 4.3, page 86). Addition (subtraction) is more involved, as subtracting two nearby numbers can cause loss of precision. In particular, the loss of precision in a₁ - a₂ due to cancellation can be estimated by using log₂ (|a₁|/|a₁ - a₂|) and log₂ (|a₂|/|a₁ - a₂|) (assumption 4.3, page 86).

Note that rounding errors are not considered in the paper. Probably, compared with the loss of precision, rounding errors are negligible in this application. Also, underflow and overflow are not considered in the paper. Underflow in particular may be of concern in this case, because it is about recognizing zero coefficients.

To move from the arithmetic operations of the &tgr; representations to the coefficients of polynomials, the author introduces the concepts of precision loss space and basis. Each element in the space is an array of the coefficients of a polynomial. Liang proves that any precision loss space has a finite weak basis (Theorem 5.9). This allows matrices to represent weak basis. Consequently, the author defines a dependence number, a measurement of the complexity of the dependence of precision losses in a precision loss space. An example in section 7 illustrates how the weak basis can be used to track precision loss in computing the Gröbner bases. A typical computation is &agr;₁ p₁ + &agr;₂ p₂, where &agr;₁ and &agr;₂ are scalars and p₁ and p₂ are polynomials. In the example on page 93, line 13, the term 0.1169543923 &tgr;¹⁰ x² should be 0.1169543923 &tgr;¹¹ x², since k_2,2 = 0 and k_2,1 = -15 in the following matrix K, thus max (15 + k_2,1, 11 + k_2,2) = 11. Also, the mix of decimal and binary can be confusing. For example, in 0.3846100000 &tgr;¹⁵, 0.3846100000 is decimal, while &tgr;¹⁵ means precision loss of 15 binary bits. That is, about the five least significant decimal digits are inaccurate.

The paper is well written. The proposed method is more precise than the previous ones. It gives more accurate estimates for precision losses and considers the undecidable cases (page 84, assumption 3.1). Potentially, the method could be integrated into a software package for the computation of Gröbner bases.