If your two polynomials are not simply constant multiples of the same thing, then you can choose a linear combination which eliminates the leading term to leave a polynomial of lower degree with a root " x " 0 .-- talk ) 18 : 53, 23 June 2016 ( UTC)
32.
The first ( greatest ) term of a polynomial for this ordering and the corresponding monomial and coefficient are respectively called the " leading term ", " leading monomial " and " leading coefficient " and denoted, in this article, lt ( ), lm ( ) and lc ( ).
33.
To count the occurrences of the individual variables in the resulting monomial, fill the column of the Young diagram corresponding to the factor concerned with the numbers of the variables, then all boxes in the first row contain 1, those in the second row 2, and so forth, which means the leading term is.
34.
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35.
Given the Gr�bner basis " G " of an ideal " I ", it has only a finite number of zeros, if and only if, for each variable " x ", " G " contains a polynomial with a leading monomial that is a power of " x " ( without any other variable appearing in the leading term ).
36.
What we know about the ? entries is that they can increment by at most 1 each time ( this is a simple argument : if f and g have the same order of pole at P, then f + cg will have a pole of lower order if the constant c is chosen to cancel the leading term ).
37.
Actually, Q wouldn't need to be quartic, and I wouldn't generally expect Q to have the same or a lower degree than P . The only thing needed is to get the same number of factors as the degree of P and for the leading terms-- coefficient and degree-- of the factors to be equal . talk ) 08 : 46, 24 January 2010 ( UTC)
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