A Harmonic Polynomial Vanishes on a Cube’s Skeleton
Harmonic functions are the quiet workhorses of potential theory: heat, gravity, and electrical fields blend into smooth, source-free shapes that obey Laplace’s equation. Their zeros—the places where the function hits zero—often feel like the hidden skeletons of these fields, revealing where a system can settle into calm equilibrium. The question of which geometric sets can be carved out as the zero set of a harmonic function has a long lineage, tracing from 19th-century curiosity to modern PDE theory. In a translation of that age-old puzzle into a modern toolkit, Ioann Vasilyev of the St. Petersburg Department of the Steklov Mathematical Institute anchors a bold claim: in any dimension d ≥ 3, you can construct a nonzero harmonic polynomial that vanishes precisely on the
d−2 skeleton of the unit cube.
The paper’s author notes that the plane behaves differently—the famous maximum principle claws back the possibility in two dimensions—but when you climb into higher dimensions the geometry becomes rich enough to permit these carefully tuned zero sets. The result is celebrated as a positive answer to a problem from the storied Scottish Book posed by R. Wavre in 1936: can a harmonic function defined in a region containing a cube vanish on all the cube’s edges without being identically zero? Vasilyev’s construction says yes, and it does so with a strikingly explicit formula.
Vanishing on the cube skeleton
At the heart of Theorem 1 is a remarkably concrete polynomial that behaves like a high-dimensional cousin of the Vandermonde determinant. Define for x = (x1, x2, …, xd) in Rd the polynomial
f_d(x) = ∏_{1 ≤ i < j ≤ d} (x_i^2 − x_j^2).
This product is the natural-looking protagonist: it vanishes whenever two coordinates hit the same magnitude, which is exactly what happens along the edges of the unit cube when you freeze most coordinates at ±1/2. Vasilyev shows that f_d is not just algebraically interesting, but harmonic: its Laplacian ∆f_d(x) equals zero for every x in Rd. In other words, a polynomial built by tallying pairwise squared differences can also be a calm, nonzero harmonic function in the whole space.
The geometric payoff is immediate: when you look at any edge of the unit cube in Rd, you fix all but two coordinates at ±1/2, and those two coordinates sit at ±1/2 as well in the right combination. In that edge, at least one pair (i, j) has x_i^2 = x_j^2, so x_i^2 − x_j^2 = 0, annihilating the whole product. Hence f_d vanishes on the d−2 skeleton of the cube—the union of all faces whose dimension is d−2. For d ≥ 3, this skeleton is nontrivial, and f_d gives a nontrivial harmonic polynomial that vanishes on it. The result is sharp in the sense that the analogous statement fails in dimension 2: the maximum principle blocks any nonzero harmonic function from vanishing on the edges of a square.
Beyond the explicit construction, the paper emphasizes a deep connection between algebraic structure and analytic behavior. The Vandermonde-like structure of f_d is not an accident; it taps into a classic theme: symmetric polynomials and their derivatives often align perfectly with the constraints of harmonicity. The proof rests on a precise auxiliary identity that ties the Laplacian to the derivatives of a Vandermonde polynomial, culminating in the simple-looking, but powerful, conclusion ∆f_d = 0. This lemma—the technical heart of the paper—lets the global geometry of the cube’s skeleton meet a locally analytic argument in a clean, constructive way.
Harmonic morphisms and higher dimensions
The second pillar of Vasilyev’s work looks beyond a single polynomial to a family of harmonics that behave in coordinated ways across dimensions. The central aim here is to produce, in every even dimension R^{2k} (k ≥ 1) and in every odd dimension R^{2k+1} (k ≥ 3), irreducible, nondegenerate homogeneous harmonic polynomials that divide infinitely many linearly independent harmonic polynomials. In plain terms: there exist “master” harmonic polynomials whose zeros become a common backbone for a whole zoo of other harmonic polynomials, across all sufficiently high dimensions. This is a powerful way to generate many nontrivial examples that share a single geometric signature.
The mechanism is the theory of harmonic morphisms, maps that preserve harmonic structure in a precise sense. Vasilyev constructs explicit harmonic morphisms in even dimensions, and then uses them to lift low-dimensional algebraic data into higher dimensions. A concrete example in the paper looks at the 2n-dimensional space with a pair of functions
ϕ1(x) = x1^2 − x2^2 + x3^2 − x4^2 + … + x_{2n−1}^2 − x_{2n}^2,
ϕ2(x) = 2(x1x2 + x3x4 + … + x_{2n−1}x_{2n}).
One of the remarkable facts borrowed from harmonic-morphism theory is that the pair (ϕ1, ϕ2) defines a harmonic morphism from R^{2n} to R^2. Then, by a classical trick in complex analysis—taking a real part of a holomorphic-like expression built from ϕ1 and ϕ2—one can generate a family of harmonic polynomials P_k that all vanish along the same set within the unit ball in Rd for all d ≥ 4. Specifically, the paper shows that for each k, the polynomial
P_k(x) = Re[(ϕ1(x) + iϕ2(x))^(2k+1)]
is harmonic, and as k grows, you obtain infinitely many linearly independent polynomials sharing the same zero set. The degree grows as 4k+2, and the leading coefficients stay delicately balanced so that the constructed polynomials remain nontrivial. This is not just a clever trick; it is a structural statement about how high-dimensional harmonic spaces encode a shared geometric core.
The result generalizes prior insights by Logunov and Malinnikova, who demonstrated similar phenomena in particular dimensions. Vasilyev’s approach, however, uses harmonic morphisms as a unifying scaffold, providing a more flexible bridge from low to high dimensions. There is a companion construction in odd dimensions (2k+1 with k ≥ 3) that mirrors the even-dimensional case, again via harmonic-morphism methods and a careful algebraic ansatz. The upshot is a robust, dimension-agnostic pathway to families of harmonic polynomials tied to the same zero set—an orbit of solutions under the action of a morphism, rather than a single, isolated polynomial.
To emphasize the more general utility, the paper also offers a lifting principle: if you have a linearly independent family of harmonic functions whose zero sets agree in a small ball, and you know of a nonconstant harmonic morphism φ, then φ ∘ f_j remains harmonic and preserves linear independence while aligning zeros with φ’s target set. In short, harmonic morphisms function like a master key that unlocks new, higher-dimensional examples from familiar, lower-dimensional seeds. This modularity is what makes the result feel expansive rather than just a collection of special-case formulas.
Open questions, reflections, and the broader picture
Even as Vasilyev builds these elegant bridges between algebra and analysis, the paper invites further exploration. In the planar setting, the author notes a natural open question: which finite unions of affine lines in Rd can sit inside the zero set of a nonzero harmonic function? The answer in the plane appears delicate: while many configurations are ruled out by reflection principles or by unique continuation, a complete classification remains tantalizing. The appendix wrestles with a concrete slice of that problem: it analyzes unions of three lines in the plane and uses Schwarz reflection principles to pin down when a nontrivial harmonic function can vanish along such a union. The upshot is both a win and a doorway—an explicit example (for a half-strip in the plane) shows nonzero harmonic functions vanishing on a boundary, while the broader classification remains open for more intricate line unions.
Another centerpiece is the construction of the explicit family of harmonic functions that vanish on the same set in the unit ball across all dimensions d ≥ 4. This demonstrates that the geometry of zero sets in harmonic functions is not a strictly dimension-bound curiosity but a structural feature that can be preserved under dimension-ascending operations. The work connects to a long thread of questions about unique continuation (the idea that vanishing on a set with enough structure forces a function to vanish identically) and to historical conjectures like Maxwell’s, which ask how the arrangement of nodal lines and surfaces constraints the behavior of physical fields. In an era where mathematicians increasingly translate between explicit algebraic constructions and geometric intuition, Vasilyev’s results feel like a rare synthesis: you get concrete polynomials, a clean analytic property (harmonicity), and a broad geometric narrative about where zero sets can live across many dimensions.
Why does it matter beyond pure math? Zero sets of harmonic functions connect to inverse problems, wave propagation, and even numerical methods that rely on constructive bases for harmonic spaces. If you can prescribe a zero set with a nonzero harmonic function, you gain a powerful tool for shaping fields and understanding how constraints propagate through space. In physics, zero sets relate to nodal lines and surfaces where wave amplitudes vanish; in engineering, they inform boundary-value problems where one seeks fields with prescribed quiet zones. Vasilyev’s explicit formulas give a language to talk about these phenomena with a clarity that invites both theoretical and applied exploration.
In closing, the work situates itself at a vibrant intersection of explicit algebraic construction, differential geometry, and partial differential equations. It shows that the geometry of a cube’s skeleton—an apparently simple, combinatorial object—can guide the assembly of globally harmonic objects in any dimension. It also demonstrates how modern tools—harmonic morphisms, lifting techniques, and structured polynomial families—can transform a classical question into a scalable framework. The author, Ioann Vasilyev, writing from the St.-Petersburg Department of the Steklov Mathematical Institute, has given readers not just a theorem, but a bridge to think about harmonic functions as carriers of geometric intent across dimensions.
Takeaways in plain language
What’s the bottom line you can carry in your pocket: first, you can design nonzero harmonic polynomials that vanish on the edges of a cube in any high dimension, and these polynomials are not random accidents but carefully crafted from a Vandermonde-like product. Second, you can take a little harmonic-morphism machinery from two variables and lift it to any higher dimension to produce a whole family of polynomials that share the same zero geometry. Third, these constructions aren’t just neat tricks; they illuminate deep structural aspects of harmonic functions, linking algebra, geometry, and analysis in a way that makes the shape of zeros feel almost programmable. And finally, they open doors to further questions about which lines or flats in the plane can be forced to disappear in the higher-dimensional harmonics, a question with roots stretching back to Maxwell and beyond.
Ioann Vasilyev’s work is a reminder that sometimes the most striking physics-style insights hide inside a comb of symbols—where a product over all pairs, a couple of squares, and a handful of clever maps can choreograph the zeros of a function across dimensions. It’s a human story of curiosity, patience, and the joy of turning a centuries-old question into a high-dimensional blueprint for the future of harmonic analysis.
