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Suppose we are given a set of $m$ polynomials $f_1,f_2,...,f_m$ in $n$ variables $z_1,...,z_n$ on an algebraically closed field, i.e. $\mathbb{C}^n[z_1,...,z_n]$. Furthermore, suppose that the set of polynomials does not share any common zeros, such that due to Hilberts weak Nullstellensatz a solution to the following equation must exist

$$Q_1 f_1 + Q_2 f_2+...+Q_m f_m=1$$

Where the $Q_i$ are certain polynomials in $\mathbb{C}^n[z_1,...,z_n]$. There exist general upper bounds on the degrees of polynomials $Q_i$. However, what I am interested in is whether there exist explicit construction methods for the $Q_i$ once an example set of $f_i$ is given? Is there a general procedure to obtain the $Q_i$? Thanks for any suggestion.

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  • $\begingroup$ What has the existence of a linear combination of $1$ to do with the weak Nullstellensatz? $\endgroup$
    – k.stm
    Oct 7, 2015 at 19:40
  • $\begingroup$ @k.stm The weak Nullstellensatz can be defined as an existence criterium in this manner. It is nicely outlined i.e. in terrytao.wordpress.com/2007/11/26/hilberts-nullstellensatz $\endgroup$
    – Kagaratsch
    Oct 7, 2015 at 19:43
  • $\begingroup$ Ok, but you literally wrote “suppose that the set of polynomials does not span a proper ideal”, which implies they span all of $ℂ[z_1, …, z_n]$, which in turn implies that $1$ is in their span which exactly says that there is a linear combination of $1$ using $f_1, …, f_m$ in the manner you stated. Don’t need any Nullstellensatz here, just unpacking definitions. Maybe you want to say “suppose that the set of polynomials doesn’t share a common zero”? Or just leave out the “due to weak Nullstellensatz” part. $\endgroup$
    – k.stm
    Oct 7, 2015 at 19:46
  • $\begingroup$ @k.stm You are right, the set of polynomials not sharing the same zero is a sufficient property for what I'd like to know. I will edit the question and change it. $\endgroup$
    – Kagaratsch
    Oct 7, 2015 at 19:52

2 Answers 2

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Yes, this can be done with Gröbner bases. I had a very similar question once. An implementation can be found in sage, which uses the lift method from singular.

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It can be done with Macaulay2 by using the // operator (line i13 below):

i7 : S=QQ[x,y]

o7 = S

o7 : PolynomialRing

i12 : id2=ideal(y-x^2,x-2, y-5)

                2
o12 = ideal (- x  + y, x - 2, y - 5)

o12 : Ideal of S

i13 : 1 // gens id2

o13 = {2} | 1   |
      {1} | x+2 |
      {1} | -1  |

              3       1
o13 : Matrix S  <--- S
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