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Sums of squares and quadratic persistence on real projective varieties. (English) Zbl 1527.14112

The authors study bounds on the Pythagoras number of real varieties \(X\): the smallest positive integer \(r\) such that any sum of squares of linear forms in the coordinate ring of \(X\) can be expressed as the sum of at most \(r\) squares.
They provide new bounds or improve known upper bounds on the Pythagoras number of varieties in terms of (algebraic) invariants of the varieties: for a real non-degenerate variety \(X\) in a projective space \(\mathbb{P}^n\),
1.
\(\binom{\text{py}(X)+1}{2}< \dim_{\mathbb{R}} R_2\), where \(R_2\) is the space of quadrics on \(X\);
2.
\(\text{py}(X) \le n+1-\min\{a(X),\text{codim}(X)\}\), where \(a(X)\) is the Green-Lazarsfeld index, i.e., the length that the minimal free resolution of \(X\) stays linear;
3.
\(\text{py}(X) \le \dim(\tilde{X}) + 1\), where \(\tilde{X}\) is any real \(2\)-regular variety containing \(X\).

In contrast, they introduce an algebraic invariant that provides a lower bound on the Pythagoras number of varieties: the quadratic persistence of a variety \(X\) is the smallest cardinality of sets \(\Gamma\) of points on \(X\) such that there is no quadrics in the defining ideal of \(\pi_\Gamma(X)\), where \(\pi_\Gamma: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-|\Gamma|}\) is the linear projection away from \(\Gamma\). They prove that, for non-degenerate \(X \subseteq \mathbb{P}^n\), \[\text{py}(X) \ge n + 1 - \text{qp}(X).\]
The authors observe that their upper and lower bounds on the Pythagoras number agree when the quadratic persistence is relatively large. They show that the quadratic persistence of a non-degenerate variety \(X\) is at most the codimension of the variety, and the equality holds if and only if the variety is a variety of minimal degree. Moreover, it is equivalent that the Pythagoras number equals the dimension of the variety plus one.
In addition, they study the next case where \(\text{qp}(X)= \text{codim}(X)-1\) for arithmetically Cohen-Macaulay variety \(X\): a variety \(X\) satisfies \(\text{qp}(X) = \text{codim}(X) - 1\) if and only if \(\text{deg}(X)= 2 + \text{codim}(X)\) or \(X\) is a codimension-one subvariety of a variety of minimal degree if and only if \(\text{py}(X) = 2 + \dim(X)\).
They also investigate properties of quadratic persistence itself, particularly in terms of the syzygies of the variety, and compute the invariants discussed in this article for several toric varieties and linear space arrangements defined by quadratic monomial ideals.

MSC:

14P05 Real algebraic sets
52A99 General convexity
13D02 Syzygies, resolutions, complexes and commutative rings

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