A Solution to Kroneckers Problem
A Solution to Kroneckers Problem
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The length of the sequence of reductions is necessarily finite, since the lexicographic or- dering on the head monomials is a well-ordering.
2. We call an irreducible A' obtained from >1 by a sequence of reductions, as above, a normal form of A with respect to M, and is not necessarily unique. We denote the set of normal forms of A by NFm(^) = {A':A-^^'}^0.
3. It is also evident that A — > A' implies that A - A' £ M, and A = 0iMi + OiAh + ■■■ + O^M, + A', where Hterm(A) > Hterm(^, A/, ), for al...l i, and Hterm(A) > Hterm(A').
lex le« A Solution to Kronecker's Problem 4. In particular, A^O (i. E. G NFm(^)) implies that A e M.
Definition 2. 3 (Property (E)) A set of generators M = {Mi, M2, ■ ■ . , Mu} of the ideal M has property {E) if AeM O A^O. In this case, we say that M is an E -basis of the ideal A1. D Note that the property (E) is equivalent to the following seemingly stronger condition: AeM ^ NFm(A) = {0}.
If, in fact, our claim were not true then there would he an A £ M\ {0}, and a choice of reduction sequence such that A — >■ A' / 0.
What to read after A Solution to Kroneckers Problem?
You can find similar books in the "Read Also" column, or choose other free books by Giovanni Gallo to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The length of the sequence of reductions is necessarily finite, since the lexicographic or- dering on the head monomials is a well-ordering.
2. We call an irreducible A' obtained from >1 by a sequence of reductions, as above, a normal form of A with respect to M, and is not necessarily unique. We denote the set of normal forms of A by NFm(^) = {A':A-^^'}^0.
3. It is also evident that A — > A' implies that A - A' £ M, and A = 0iMi + OiAh + ■■■ + O^M, + A', where Hterm(A) > Hterm(^, A/, ), for al...l i, and Hterm(A) > Hterm(A').
lex le« A Solution to Kronecker's Problem 4. In particular, A^O (i. E. G NFm(^)) implies that A e M.
Definition 2. 3 (Property (E)) A set of generators M = {Mi, M2, ■ ■ . , Mu} of the ideal M has property {E) if AeM O A^O. In this case, we say that M is an E -basis of the ideal A1. D Note that the property (E) is equivalent to the following seemingly stronger condition: AeM ^ NFm(A) = {0}.
If, in fact, our claim were not true then there would he an A £ M\ {0}, and a choice of reduction sequence such that A — >■ A' / 0.
What to read after A Solution to Kroneckers Problem?
You can find similar books in the "Read Also" column, or choose other free books by Giovanni Gallo to read onlineMoreLess
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