Resultant

In mathematics , the resultant is a commutative algebra tool to check two polynomials for the presence of common zeros. As an extension to multivariate polynomial systems of equations, the resultant can be used to successively eliminate the variables of the system. For this purpose the resultant and similar constructions were investigated in the course of the 19th century, first for systems with symmetries, then in 1882 by L. Kronecker also for the general case. In modern computer algebra systems , resultants or their multidimensional analogues are used to infer the solutions (or their approximations) of an equation system from a previously determined Gröbner basis .

definition

Let and two polynomials of degree or from , the polynomial ring in an indeterminate over a commutative unitary ring , written out ${\ displaystyle f}$ ${\ displaystyle g}$ ${\ displaystyle m}$ ${\ displaystyle n}$ ${\ displaystyle R [X]}$ ${\ displaystyle X}$ ${\ displaystyle R}$

{\ displaystyle f = f_ {0} + f_ {1} X + \ dotsb + f_ {m} X ^ {m}}

and .

{\ displaystyle g = g_ {0} + g_ {1} X + \ dotsb + g_ {n} X ^ {n}}

The resultant of these two polynomials is the determinant of the New Year's Eve matrix .

{\ displaystyle \ operatorname {Res} (f, g) = \ det {\ begin {pmatrix} f_ {m} & f_ {m-1} & \ cdots && f_ {0} &&& \\ & f_ {m} & f_ {m- 1} & \ cdots && f_ {0} && \\ && \ ddots &&&& \ ddots & \\ &&& f_ {m} & f_ {m-1} & \ cdots && f_ {0} \\ g_ {n} & g_ {n-1} & \ cdots && g_ {0} &&& \\ & g_ {n} & g_ {n-1} & \ cdots && g_ {0} && \\ && \ ddots &&&& \ ddots & \\ &&& g_ {n} & g_ {n-1} & \ cdots && g_ {0} \\\ end {pmatrix}}}

The matrix consists of rows with the coefficients of and rows with the coefficients of . All entries not labeled in the above matrix are zero. The New Year's Eve matrix is therefore a square matrix with rows and columns. ${\ displaystyle n}$ ${\ displaystyle f}$ ${\ displaystyle m}$ ${\ displaystyle g}$ ${\ displaystyle m + n}$

properties

The ( transpose of the) New Year's Eve matrix is the system matrix of the equation , understood as a linear system of equations in the coefficients of the cofactor polynomials ${\ displaystyle fp-gq = 0}$

{\ displaystyle p = p_ {0} + p_ {1} X + \ dotsb + p_ {n-1} X ^ {n-1}}

and .

{\ displaystyle q = q_ {0} + q_ {1} X + \ dotsb + q_ {m-1} X ^ {m-1}}

If the polynomials and have a common factor, the resultant vanishes. For the statement in the other direction one still needs that the ring has a factorial integrity domain , i. H. without a zero divisor and with a unique prime factorization . This is always the case when there is a body , e.g. B. the field of the rational or real numbers or a polynomial ring above it. If these conditions are met and is so contained and a common factor with positive degrees. ${\ displaystyle f}$ ${\ displaystyle g}$ ${\ displaystyle R}$ ${\ displaystyle R}$ ${\ displaystyle \ operatorname {Res} (f, g) = 0}$ ${\ displaystyle f}$ ${\ displaystyle g}$

If the range of coefficients is an algebraically closed field, like the field of complex numbers, the polynomials and decompose into linear factors ${\ displaystyle f}$ ${\ displaystyle g}$

{\ displaystyle f (X) = f_ {m} \ cdot (X-a_ {1}) \ dotsm (X-a_ {m})}

and .

{\ displaystyle g (X) = g_ {n} \ cdot (X-b_ {1}) \ dotsm (X-b_ {n})}

In this case the resultant can be represented as an expression in the zeros, it applies

{\ displaystyle \ operatorname {Res} (f, g) = (f_ {m}) ^ {n} \, g (a_ {1}) \ dotsm g (a_ {m}) = (- 1) ^ {mn } (g_ {n}) ^ {m} \, f (b_ {1}) \ dotsm f (b_ {n}) = (f_ {m}) ^ {n} \, (g_ {n}) ^ { m} \ prod _ {i = 1} ^ {m} \ prod _ {j = 1} ^ {n} (a_ {i} -b_ {j})}

.

With the help of Cramer's rule one can show that there are always polynomials and with coefficients in such that ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle R}$

{\ displaystyle Af + Bg = \ operatorname {Res} (f, g)}

applies. The coefficients of and result from the last column of the complementary matrix of the New Year's Eve matrix. ${\ displaystyle A}$ ${\ displaystyle B}$

Relationship to the Euclidean Algorithm

A similar formula is obtained using the extended Euclidean algorithm . Indeed, an efficient calculation method for the resultant can be derived from this, the sub-resultant method.

literature

Siegfried Bosch : Algebra. 7th, revised edition. Springer, Berlin et al. 2009, ISBN 978-3-540-92811-9 , doi : 10.1007 / 978-3-540-92812-6 .