Necessity and minimal regularity hypotheses in Douglas's converse theorem on the diametral locus of an algebraic curve

Douglas (1940) proved a converse to Newton's diameter theorem: if the centroid of the $n$ intersection points of $n$ plane curve-elements with a translating secant line traces a straight line for every system of parallel secants, then the elements all belong to one possibly reducible algebraic curve of degree $n$ . His proof, however, stipulates regularity: the branch functions describing the elements must admit derivatives through order $n$ (real domain) or the elements must be analytic in small regions (complex domain). Section 7 of the same paper raises but does not resolve whether such hypotheses are needed. For $n = 2$ , in the particular form he treats, Douglas shows mere continuity suffices, the condition forcing a vanishing third difference and hence a parabola or two-fold straight line; for $n > 2$ , the necessity of the smoothness assumptions, the minimal hypotheses under which the theorem holds, and the behavior under restricted families of secant directions are all left undetermined. The hypothesis-strength question is open.

Background

An algebraic curve of degree $n$ in the plane is the solution set of a polynomial equation $F(x,y) = 0$ with $F$ of total degree $n$ ; the curve is called reducible if $F$ factors into polynomials of lower degree. A classical theorem of Newton states that if a line meets such a curve in $n$ points, counted with multiplicity, then the centroid (arithmetic mean) of those $n$ points lies on a fixed straight line, called a diameter of the curve, as the line translates parallel to itself. Jesse Douglas (1940) proved a converse to this theorem. A curve-element is a small arc of a plane curve, small enough that a generic line meets it in a single point, and Douglas works with $n$ such elements. Fix a direction and consider the family of all lines parallel to it; each line of the family is a secant of the configuration. The centroid of the $n$ points where one secant meets the $n$ elements traces, as the secant translates, the diametral locus of the elements relative to that direction. Douglas's Theorem II states that if the diametral locus is a straight line for every system of parallel secants, then the $n$ elements all belong to one possibly reducible algebraic curve of degree $n$ . Theorem I gives a stronger conclusion under a more restrictive hypothesis: if all the diametral lines share one invariable direction $ax+by = \mathrm{const.}$ — the restriction under which Levi's analytic theorem reappears as a special case — then the curve has the special form $(ax+by)^n + R_{n-1}(x,y) = 0$ with $R_{n-1}$ of degree at most $n-1$ . Theorem III extends Theorem II to hypersurface-elements in Euclidean spaces of any finite dimension, the diametral locus being a hyperplane. The proofs require regularity hypotheses: in the real domain Douglas supposes that the branch functions $g_i$ describing the elements as graphs admit derivatives through order $n$ (in his notation, $\partial^j g_i/\partial c^j$ exists for $j = n, n-1, \ldots, 1$ ), and in the complex domain the elements are assumed analytic in sufficiently small regions. In Section 7 he observes that for $n = 2$ no such hypothesis is needed in the particular form he treats: for a merely continuous curve $y = f(x)$ , the diametral condition forces the third difference of $f$ (the combination $f(x+3h) - 3f(x+2h) + 3f(x+h) - f(x)$ on four equally spaced points) to vanish for all arguments in arithmetic progression, and a continuous function with vanishing third differences is a parabola, in degenerate cases a straight line possibly counted twice as a non-reduced degree-2 curve. The question descends from an analytic precursor of Howard Levi (1939): if $f$ is an entire function (complex-analytic everywhere in the plane) and the $n$ branches $h_i(x,k)$ of the difference-quotient equation $[f(x+h) - f(x)]/h = k$ are analytic, distinct, nonzero, and satisfy $\sum \partial h_i/\partial x = -(n+1)$ , then $f$ is a polynomial of degree $n+1$ . Douglas's Theorem II is the geometric generalization of Levi's result; Levi's hypotheses are analytic throughout, and the question of weaker regularity does not arise in his paper.

Problem Statement

For $n \geq 3$ , determine the role of the regularity hypotheses in Douglas's converse theorem on the diametral locus. The theorem's conclusion is that $n$ curve-elements whose diametral locus is a straight line for every system of parallel secants belong to one possibly reducible algebraic curve of degree $n$ ; its proof currently stipulates that the branch functions $g_i$ admit derivatives through order $n$ in the real domain, or analyticity of the elements in the complex domain. The open questions, raised in Section 7 of Douglas (1940), are: (1) Necessity — does the hypothesis of the theorem, namely that the diametral locus is a straight line for every system of parallel secants, itself force the regularity the proof consumes, so that the theorem remains valid for merely continuous curve-elements, as Douglas verified for the particular form of the case $n = 2$ that he treats? Or does there exist, for some $n \geq 3$ , a configuration of $n$ continuous curve-elements, each met by a generic secant in a single point, whose diametral locus is a straight line for every direction, yet which do not all belong to any possibly reducible algebraic curve of degree $n$ ? (2) Minimal hypotheses — independently of the extreme case of continuity, identify the weakest regularity class or structural hypotheses under which Theorem II for curves and Theorem III for hypersurfaces remain valid. (3) Restricted directions — Theorem II demands the straight-line diametral property for every system of parallel secants; determine for which restricted families of secant directions the characterization survives and what form it then takes. The case $n = 2$ is settled in the source, in the particular form Douglas treats, and constrains any answer: the resulting curve is a parabola or a two-fold straight line. The three facets are logically independent; each is posed with a definite class of hypotheses and a definite range of $n$ and dimension.

References

Jesse Douglas, "A converse theorem concerning the diametral locus of an algebraic curve", Duke Mathematical Journal 6 (1940), no. 2, pp. 375-388. DOI: 10.1215/s0012-7094-40-00629-9. URL: https://doi.org/10.1215/s0012-7094-40-00629-9
Howard Levi, "On the values assumed by polynomials", Bulletin of the American Mathematical Society 45 (1939), no. 8, pp. 570-575. DOI: 10.1090/S0002-9904-1939-07038-9. URL: https://doi.org/10.1090/S0002-9904-1939-07038-9 (full text available at https://www.ams.org/journals/bull/1939-45-08/S0002-9904-1939-07038-9/S0002-9904-1939-07038-9.pdf)