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My question is assume that we know that the degree of some irreducible variety is small does it possible to conclude that there exists polynomial of small degree vanishing on this variety.

Let us make the question more concrete: Let $V\subset A^{2n}$ be an irreducible algebraic variety of dimension $n$ and degree $d$ (say $2^n$). Let $I \triangleleft \mathbb{C}[x_1,\ldots, x_{2n}]$ be an ideal of all polynomials vanishing on $I$. My question is what is the best upper bound on the minimal degree of polynomials in $I$.

The best upper bound that I know is $2^n$, while I think that it is not the true. For example if $V$ is a complete intersection variety then $I$ must have polynomial of degree $2$.

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Probably in this question co-dimension of the variety is more important than its dimension, but for simplicity let us assume that both of them $n$. –  Klim Efremenko Jul 25 '12 at 7:32
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You might want to look into the concept of a Gröbner basis. See Eisenbud, "Commutative Algebra with a view...", chap. 15. –  Damian Rössler Jul 25 '12 at 8:05
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If you project $V$ generically onto $\mathbb A^{n+1}$, the image is an hypersurface $W$. This gives you a degree $d$ equation (in $n+1$ variables) that vanishes on $V$. –  rita Jul 26 '12 at 8:33
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2 Answers

up vote 2 down vote accepted

The minimal degree of a polynomial that vanishes on $V$ is the minimal $m$ such that $h_I(m)\ne 0$, where $h_I$ denotes the Hilbert function of the variety $V$. Hence, you are interested in upper (and lower?) bounds on the Hilbert function of certain varieties. There are two papers I know that deal with such bounds:

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It is not really an answer, but since bounty ends soon I will accept it. –  Klim Efremenko Aug 2 '12 at 11:57
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You can have a look at the article "Direct methods for primary decomposition" by Eisenbud, Huneke and Vasconselos (Inventiones 110, 1992), available on Eisenbud's webpage. In proposition 3.5, they prove (and say it is long known) that a homogeneous equidimensional ideal $I$ of degree $d$ is generated up to radical by forms of degree at most $d$.

See also the remark following the proposition for an example showing it is sharp: $V$ is the union of $d$ skew lines that all meet a common line $L$. They also mention a conjecture in the case where $I$ is prime: up to a radical it should be generated by form of degree at most $d-$ codim $I + 1$.

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