algebraically independent

algebraically independent

[¦al·jə¦brā·ik·lē ‚in·də′pen·dənt]
(mathematics)
A subset S of a commutative ring B is said to be algebraically independent over a subring A of B (or the elements of S are said to be algebraically independent over A) if, whenever a polynominal in elements of S, with coefficients in A, is equal to 0, then all the coefficients in the polynomial equal 0.
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We say that two elements a and b of an algebra B are algebraically independent if P [member of] C[[z.
2] are two algebraically independent functions, then the algebra generated by them is isomorphic to C[[z.
n]) if, adding any number of other indeterminates (elements transcendental and algebraically independent over K([X.
n+r] are transcendental and algebraically independent over K([X.
16] are algebraically independent over F and hence the polynomial Q is uniquely determined.
Let O be an open set in K, and let P [member of] O be a polytope with vertex set V which we suppose to be in general position, for instance the coordinates of the vertices are algebraically independent.
Another approach is to assign algebraically independent values to the nonzeros (i.
T] (the latter graph is equivalent to the directed graph of F with the edge directions reversed); this equivalence again assumes that the numerical values of the nonzeros in F are algebraically independent.
Y,y] is algebraically independent of the cycles [Z.
r] : r [member of] H} is algebraically independent.
1 Any maximal weakly separated subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] corresponds to k(n - k) +1 algebraically independent Plucker coordinates [[DELTA].
W] is again a polynomial algebra, and it can be generated by n algebraically independent homogeneous polynomials [f.