Proposition

Rewrite each of the following polynomials, ordering the terms using the lex order, grlex order, and the grevlex order, giving $\LM(f), \LT(f)$ and $\multideg(f)$ in each case.

1. $f(x, y, z) = 2x + 3y + z + x^2 - z^2 + x^3$.
2. $f(x, y, z) = 2x^2y^8 - 3x^5yz^4 + xyz^3 - xy^4$.

Solution

1

• lex: $x^3 + x^2 + 2x + 3y - z^2 + z$.
  • $\LM(f) = x^3$.
  • $\LT(f) = x^3$.
  • $\multideg(f) = 3$.
• grlex: $x^3 + x^2 - z^2 + 2x - 3y + z$.
  • $\LM(f) = x^3$.
  • $\LT(f) = x^3$.
  • $\multideg(f) = 3$.
• grevlex: $x^3 - z^2 + x^2 + z - 3y + 2x$.
  • $\LM(f) = x^3$.
  • $\LT(f) = x^3$.
  • $\multideg(f) = 3$.

2

• lex: $-3x^5yz^4 + 2x^2y^8 - xy^4 + xyz^3$.
  • $\LM(f) = x^5yz^4$.
  • $\LT(f) = -3x^5yz^4$.
  • $\multideg(f) = 10$.
• grlex: $-3x^5yz^4 + 2x^2y^8 - xy^4 + xyz^3$.
  • $\LM(f) = x^5yz^4$.
  • $\LT(f) = -3x^5yz^4$.
  • $\multideg(f) = 10$.
• grevlex: $2x^2y^8 - 3x^5yz^4 - xy^4 + xyz^3$.
  • $\LM(f) = x^2y^8$.
  • $\LT(f) = 2x^2y^8$.
  • $\multideg(f) = 10$.