Let M be an (mx n)-matrix with nonnegative real entries and let r, 12,...,m denote the...

 

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Let M be an (mx n)-matrix with nonnegative real entries and let r, 12,...,m denote the rows of M. Assume that ker(M) nZ" = {0}. Define a binary relation >M on the monomials in the polynomial ring S = K[x1,x2,...,xn] as follows: x">M x if there is an positive integer i (at most m) such that u r;= v r; for all 1 M is a monomial order on the polynomial ring S. (ii) When M:=ii [], show that >M equals > grevlex on K[x, y, z). (iii) For the (nx n)-identity matrix I, show that >lex equals >1. Let M be an (mx n)-matrix with nonnegative real entries and let r, 12,...,m denote the rows of M. Assume that ker(M) nZ" = {0}. Define a binary relation >M on the monomials in the polynomial ring S = K[x1,x2,...,xn] as follows: x">M x if there is an positive integer i (at most m) such that u r;= v r; for all 1 M is a monomial order on the polynomial ring S. (ii) When M:=ii [], show that >M equals > grevlex on K[x, y, z). (iii) For the (nx n)-identity matrix I, show that >lex equals >1.

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To prove the given statements lets break down the problem into three parts i Show that is a monomial order on the polynomial ring S To demonstrate that is a monomial order we need to establish three p... View the full answer