Definitions for leading monomials, coefficients, and terms - Maple Help

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Definitions for leading monomials, coefficients, and terms

 

Description

Examples

Description

• 

The current terminology used in Groebner is that of Ideals, Varieties and Algorithms by David Cox, John Little, and Donal O'Shea, Springer-Verlag, (1992).  In releases of Maple prior to Maple 10, Groebner used a different, conflicting convention. In particular, what is now called a leading monomial used to be called a leading term, and vice versa.

• 

Note: the old commands Groebner[leadmon] and Groebner[leadterm] still respect the old convention, however they are now deprecated.  You should replace them with Groebner[LeadingTerm] or Groebner[LeadingMonomial] respectively.

• 

The current convention is as follows:

– 

A monomial is a product of indeterminates from a fixed set X, possibly with repetitions. The coefficient of a monomial is always one.

– 

A term of a polynomial (with respect to X) is the product of a monomial in X and a coefficient whose degree in X is zero. This coefficient may include other indeterminates not in X.  For example, the coefficients may be rational functions in other variables.

– 

The "leading term" of a polynomial with respect to a monomial order is the term whose monomial is greatest with respect to the order and whose coefficient is non-zero.  The coefficient and monomial of this term are called the "leading coefficient" and "leading monomial" of the polynomial, respectively.

• 

Note that the LeadingTerm command does not actually output terms, but rather the sequence (leading coefficient, leading monomial). This may be changed in a future release of Maple.

Examples

withGroebner:

f5x3y+x2w2t+5x3yzt2xzw3t+3y2w3t

f:=2tw3xz+3tw3y2+5tx3yz+tw2x2+5x3y

(1)

With respect to the definitions above, we will compute the leading coefficient, leading monomial, and leading term of the polynomial f with respect to lexicographic order with x > y > z > w > t.

LeadingCoefficientf,plexx,y,z,w,t

5

(2)

LeadingMonomialf,plexx,y,z,w,t

x3yzt

(3)

`*`LeadingTermf,plexx,y,z,w,t

5x3yzt

(4)

In releases of Maple prior to Maple 10, Groebner[leadmon] computed what is now returned by LeadingTerm and Groebner[leadterm] computed what is now returned by LeadingMonomial.

leadcoefff,plexx,y,z,w,t

Warning, Groebner[leadcoeff] is deprecated. Please, use Groebner[LeadingCoefficient].

5

(5)

leadtermf,plexx,y,z,w,t

Warning, Groebner[leadterm] is deprecated. Please, use Groebner[LeadingMonomial]. See ?Groebner,terminology for details.

x3yzt

(6)

`*`leadmonf,plexx,y,z,w,t

Warning, Groebner[leadmon] is deprecated. Please, use Groebner[LeadingTerm]. See ?Groebner,terminology for details.

5x3yzt

(7)

See Also

LeadingTerm

MonomialOrders

 


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