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This is likely a duplicate, but I was unable to find a reference, so apologies if it is.

In the case of a polynomial with one variable, say $f=a_nX^n + ... + a_1X + a_0$, I would say the common definition of the leading term of $f$ is the term $a_nX^n$. Fixing a monomial ordering we can generalize this definition to multivariate polynomials, as is suggested for example here.

Yet, when reading about tangent cones in algebraic geometry, another notion of leading term arises: Any multivariate polynomial $f$ of degree $n$ has a unique decomposition $f = \sum_{k=0}^n f^{(k)}$ into homogeneous polynomials of degree $k=0...n$. The leading term of $f$ is the first (minimal $k$) nonzero $f^{(k)}$. This appears for example in Eisenbud Harris - Geometry of Schemes (p. 107) or Görtz Wedhorn - Algebraic Geometry I (p. 166)

Does the latter notion have another name? What is a good resource to read about it?

As ideals generated by the latter notion seem to be relevant, I would like to find more information, but the former notion makes searching for it rather difficult.

As always thank you very much for your time.

Jonas Linssen
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    Really, the leading term just depends on the ordering you choose. Anyway, the leading term is sometimes also called the initial term, as in this paper by Sturmfels. Searching for "initial ideal" gives lots of results. – Viktor Vaughn Jan 06 '21 at 18:41
  • Right, but what is that thing Eisenbud, Harris, Görtz & Wedhorn talk about? – Jonas Linssen Jan 06 '21 at 21:18
  • Ah, I see. In this case it's called the initial form, as in this paper. Two other references are Cox, Little, O'Shea's Using Algebraic Geometry section 8.5 and Eisenbud's Commutative Algebra... sections 5.1 and 7.5. There is also the term local order but I think it's not quite what you're after, since most authors still usually define this as a monomial order. – Viktor Vaughn Jan 07 '21 at 01:30
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    Thank you very much, this is really helpful! – Jonas Linssen Jan 07 '21 at 14:25

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