Publisher's Foreword.- Editors' Foreword.- Introduction.- 2 Geometry of Conic Sections.- 3 The Physics of Conic Sections and Ellipsoids.- 4 Projective Geometry.- 5 Complex Algebraic Curves.- 6 A Problem for School Pupils.- A Into How Many Parts do n Lines Divide the Plane?- Editors' Comments on Gudkov's Conjecture.- Notes

Real Algebraic Geometry

This paper gives a survey, with detailed references to the literature, on recent developments in real algebra and geometry concerning the polarity between positivity and sums of squares. After a review of foundational material, the topics covered are Positiv- and Nichtnegativstellensatze, local rings, Pythagoras numbers, and applications to moment problems.

/pdf/positivity-and-sums-of-squares-a-guide-to-recent-results-130nx7l8mx.pdf

Positivity and sums of squares: A guide to recent results

We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions with rational coefficients. In the case of ternary quartics, we prove that our counterexamples to Sturmfels' question are the only ones.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=18&iss=7&rank=3

Sums of squares of polynomials with rational coefficients

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal representation. Br\"and\'en has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. So the generalized Lax conjecture fails badly. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation, improving upon Br\"and\'en's mostly unconstructive result. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence.

Polynomials with and without determinantal representations

Let W be a subset of the set of real points of a real algebraic variety X. We investigate which functions $$f: W \rightarrow \mathbb {R}$$
 are the restrictions of rational functions on X. We introduce two new notions: curve-rational functions (i.e., continuous rational on algebraic curves) and arc-rational functions (i.e., continuous rational on arcs of algebraic curves). We prove that under mild assumptions the following classes of functions coincide: continuous hereditarily rational (introduced recently by the first named author), curve-rational and arc-rational. In particular, if W is semialgebraic and f is arc-rational, then f is continuous and semialgebraic. We also show that an arc-rational function defined on an open set is arc-analytic (i.e., analytic on analytic arcs). Furthermore, we study rational functions on products of varieties. As an application we obtain a characterization of regular functions. Finally, we get analogous results in the framework of complex algebraic varieties.

Curve-rational functions

Pythagoras Numbers of Real Irreducible Algebroid Curves and Gram Matrices

We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.

https://hal.archives-ouvertes.fr/hal-01894175/document

Integral closures in real algebraic geometry

Let K be a totally real Galois number field. Hillar proved that if f ∈ ℚ[x1, ..., xn] is a sum of m squares in K[x1, ..., xn], then f is a sum of N(m) squares in ℚ[x1, ..., xn], where N(m) ≤ 2[K:ℚ]+1 · $ \left( {_2^{[K:\mathbb{Q}] + 1} } \right) $ · 4m. We show in fact that N(m) ≤ m + 4$ \left\lceil {\tfrac{m} {4}} \right\rceil $ ([K: ℚ] − 1), our proof being constructive too. Moreover, we give some examples where this bound is sharp, for instance in the case of quadratic extensions. We also extend our results to the setting of non-commutative polynomials over ℚ.

https://hal.archives-ouvertes.fr/hal-00403920/document

Tight bounds for rational sums of squares over totally real fields

We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions on real algebraic varieties. We provide in particular several characterizations (algebraic or geometric) of these varieties, and provide a full description of centrally seminormal curves.

Weak and semi normalization in real algebraic geometry

We give some examples of trace-positive non-commutative quaternary quartics which are not cyclically equivalent to a sum of hermitian squares. Since some similar examples of ternary sextics were already known, this settles a perfect analogy to Hilbert's results from the commutative context which says that in general, positive (commutative) polynomials are not necessarily sums of squares, the first non trivial cases being obtained for ternary sextics and quaternary quartics.

/pdf/trace-positive-non-commutative-polynomials-3ak8nc0nv6.pdf

trace-positive non-commutative polynomials

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