Polynomial expressions

 

The syntax for polynomial expressions is similar to integer expressions.

The legal operators for polynomials, in order of increasing precedence, are given in the following table.

• +, -
• *, / (polynomial division), & (polynomial remainder)
• +, - unary operators
• ^, ** (both are exponentiation operators)

The polynomial division and remainder are done using the defined monomial order in the base ring. Parentheses are used to enforce groupings in an expression.

The following remarks describe certain aspects of polynomial expressions.

• Polynomials in Macaulay are output in an abbreviated notation: each polynomial is displayed as a number of monomials separated by ``+'' or ``-''. Each monomial is preceeded by its coefficient and each indeterminate is followed by its degree with no intervening blanks. Rational number coefficients are represented by x/y. A polynomial which extends over a line boundary is displayed by using as the continuation character. If you wish to have polynomials output in Mathematica format, use the prmat command.
• User defined matrix variables can appear in polynomial expressions. The variable must be enclosed in curly braces so that Macaulay won't think that it is a sequence of ring indeterminates. ``{M}'' refers to the (1,1) entry of the matrix M.
• A at the end of a line, preceded by a space, is the line continuation character. Macaulay computes the polynomial on each continued line, and then adds them all together. Thus, each line must be a well-defined polynomial.
• Certain abbreviations are allowed, mainly in order to allow easier entry of polynomials, and to allow the output polynomials of Macaulay to be legal input. One can suppress the ``*'' between a number or a ring indeterminate and another ring indeterminate, the ``^'' between a ring indeterminate and its exponent, and the ``*'' before a left parenthesis, or after a right parenthesis.

For example,

%  poly f (x[1]+x[2])^2

%  type f
; x[1]2+2x[1]x[2]+x[2]2 

%  int i 1

%  poly g (x[i]-x[i+1])^2*{f}

%  type g
; x[1]4-2x[1]2x[2]2+x[2]4 

%  poly h ab2+c2d+a(b+c)(b-c)

%  type h
; 2ab2-ac2+c2d