Lissajous Discriminants in Algebraic Geometry

• Lissajous discriminants are defined as the branch locus of projections from incidence varieties, capturing singular fibers, solution mergers, and bifurcations.
• They are computable via determinantal equations with degree bounds and irreducibility criteria influenced by geometric symmetry and arithmetic conditions such as coprimality.
• The concept spans applications from dynamical systems and quantum state selection to knot theory, where discriminant-like criteria distinguish degenerate regimes from stable configurations.

“Lissajous discriminants” denotes a precise object in recent algebraic geometry and a broader family of discriminant-like criteria across the Lissajous literature. In the formal sense, the term refers to the branch locus of the projection from an incidence variety associated with a Lissajous variety to parameter space; this locus records singular fibers, solution mergers, and bifurcations (Mascarin et al., 8 Sep 2025). In earlier and parallel settings, the same expression is used more loosely for arithmetic, geometric, spectral, or dynamical features that distinguish Lissajous families, including irreducibility of Chebyshev-Lissajous polynomials, singular loci of Chebyshev varieties, nondegeneracy conditions for Lissajous shadows of knots, quantum pattern selection, prime-frequency asymmetry, and nonlinear resonance manifolds (Zhang, 2022, Dencker et al., 2017, Soret et al., 2015, Russo, 2024, Barna et al., 2020, Escobar-Ruiz et al., 23 Jun 2026).

1. Formal definition from Lissajous varieties

The most explicit definition of a Lissajous discriminant arises in the theory of Lissajous varieties. Starting with a rational matrix

$A\in \mathbb{Q}^{d\times n},$

and a shift vector $b\in\mathbb{C}^n$, one defines the affine subspace

$$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$

and its coordinatewise cosine image

$${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$

With a parametrization of $\mathrm{Row}(A)$ by $t\in\mathbb{C}^d$, this becomes

$$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$

where $a_j$ is the $j$-th column of $A$. After setting $b\in\mathbb{C}^n$0 and $b\in\mathbb{C}^n$1, the same map admits the Laurent form

$b\in\mathbb{C}^n$2

Special cases are $b\in\mathbb{C}^n$3, giving $b\in\mathbb{C}^n$4, and $b\in\mathbb{C}^n$5, giving $b\in\mathbb{C}^n$6 (Mascarin et al., 8 Sep 2025).

The foundational geometric facts are that $b\in\mathbb{C}^n$7 is closed in $b\in\mathbb{C}^n$8, irreducible, and satisfies

$b\in\mathbb{C}^n$9

The paper also relates $$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$0 to a scaled toric variety through an auxiliary variety $$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$1, so the ostensibly transcendental cosine image becomes an affine algebraic variety (Mascarin et al., 8 Sep 2025).

Within this framework, the Lissajous discriminant is defined from the incidence variety

$$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$2

Its toric Jacobian matrix is

$$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$3

The ramification locus is

$$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$4

and the Lissajous discriminant is its image in parameter space,

$$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$5

If $$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$6 is a hypersurface, its defining polynomial is denoted $$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$7. Geometrically, $$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$8 is the set of parameter values $$L_{A,b}=\mathrm{Row}(A)-\frac{b\pi}{2}\subseteq \mathbb{C}^n,$$9 for which the fiber $${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$0 becomes singular: solutions collide, Jacobian rank drops, or the number of real equilibria changes. In the real dynamical system considered in the same work, it is exactly the bifurcation locus (Mascarin et al., 8 Sep 2025).

2. Degree, determinantal equations, and dynamical bifurcation

The geometry of a Lissajous variety is controlled by the centrally symmetric polytope

$${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$1

For generic $${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$2,

$${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$3

where $${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$4 is the number of coloops of $${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$5. More generally,

$${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$6

Thus the degree is a volume divided by the lattice index and the generic fiber degree of the projection $${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$7 (Mascarin et al., 8 Sep 2025).

The same paper derives defining equations from rank conditions. For Laurent binomial generators $${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$8 of the toric ideal of the associated toric variety, one forms matrices $${\cal L}_{A,b}=\cos(L_{A,b})=\{(\cos(x_1),\dots,\cos(x_n)):x\in L_{A,b}\}.$$9 in the quotient

$\mathrm{Row}(A)$0

Then

$\mathrm{Row}(A)$1

When the toric variety is a hypersurface, this reduces to a single determinant equation,

$\mathrm{Row}(A)$2

These determinantal descriptions make the Lissajous discriminant computable by elimination: $\mathrm{Row}(A)$3

The principal dynamical motivation is the Kuramoto model,

$\mathrm{Row}(A)$4

With the substitution $\mathrm{Row}(A)$5, $\mathrm{Row}(A)$6, the steady-state equations become polynomial, and for the reduced incidence matrix $\mathrm{Row}(A)$7 of a graph the equilibria satisfy

$\mathrm{Row}(A)$8

Accordingly, steady states are intersections of a linear space with a sine-type Lissajous variety, and $\mathrm{Row}(A)$9 gives an upper bound on the number of isolated equilibria. The positive region

$t\in\mathbb{C}^d$0

admits a convex-optimization characterization: a point $t\in\mathbb{C}^d$1 is exactly the unique minimizer of a strictly convex program, and the minimizer corresponds to a linearly stable equilibrium because the Jacobian is negative definite on the positive branch (Mascarin et al., 8 Sep 2025).

The same structure persists for the generalized system

$t\in\mathbb{C}^d$2

whose steady states satisfy $t\in\mathbb{C}^d$3, equivalently $t\in\mathbb{C}^d$4. The positive branch $t\in\mathbb{C}^d$5 and its image $t\in\mathbb{C}^d$6 are again characterized by a strictly convex optimization problem; a minimizer exists if and only if $t\in\mathbb{C}^d$7, and when it exists it is unique and yields a stable equilibrium (Mascarin et al., 8 Sep 2025).

For the discriminant itself, the paper proves the degree bound

$t\in\mathbb{C}^d$8

when the discriminant is a hypersurface. It also records explicit examples. For $t\in\mathbb{C}^d$9, the discriminant is a univariate polynomial in $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$0, with roots corresponding to tangencies of a line with the Lissajous curve. For the $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$1 graph example, the discriminant curves $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$2 and $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$3 have degrees $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$4 and $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$5, respectively, and partition parameter space into chambers with different numbers of real solutions. The discriminant further inherits symmetries: for $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$6,

$$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$7

and if $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$8 comes from a graph $$\phi_{A,b}(t)=\big(\cos(a_1\cdot t-b_1\tfrac{\pi}{2}),\dots,\cos(a_n\cdot t-b_n\tfrac{\pi}{2})\big),$$9, then

$a_j$0

for every $a_j$1 (Mascarin et al., 8 Sep 2025).

3. Chebyshev-Lissajous polynomials and arithmetic irreducibility

Before the formal branch-locus definition, a major algebraic line of work treated “Lissajous discriminants” through parameter elimination and factorization. A classical Lissajous curve is written as

$a_j$2

with $a_j$3 coprime and $a_j$4. Setting

$a_j$5

and using Chebyshev polynomials $a_j$6, defined by

$a_j$7

one obtains

$a_j$8

If $a_j$9, eliminating the parameter yields

$j$0

This is the basic Chebyshev-Lissajous polynomial, whose zero set is an algebraic model for the underlying Lissajous curve (Zhang, 2022).

The decisive structural criterion is the arithmetic of $j$1. In the nondegenerate case $j$2, the paper confirms Merino’s conjectures and proves that

$j$3

is irreducible over $j$4 if and only if $j$5, and in that coprime case it is also irreducible over $j$6. Thus coprimality is both necessary and sufficient for irreducibility over $j$7, and sufficient for irreducibility over $j$8 (Zhang, 2022).

When $j$9, the same polynomial factors as

$A$0

where $A$1 and $A$2. Each factor has coprime indices $A$3, so each factor defines an irreducible Lissajous-type curve, and the original curve becomes a finite union of such curves. In this algebraic setting, the arithmetic datum $A$4 functions as the central classifier of reducibility and geometric decomposition (Zhang, 2022).

The equal-index case admits an explicit factorization into conics: $A$5 Geometrically, this says that the equal-index curve is reducible and decomposes into a finite union of ellipses or circles, depending on the angle parameters. The special case $A$6 yields the equal-frequency polynomial $A$7 (Zhang, 2022).

The degenerate case $A$8 collapses to $A$9. Here the classification is subtler:

• $b\in\mathbb{C}^n$00 is irreducible over $b\in\mathbb{C}^n$01 if and only if $b\in\mathbb{C}^n$02.
• $b\in\mathbb{C}^n$03 is irreducible over $b\in\mathbb{C}^n$04 if and only if $b\in\mathbb{C}^n$05.

These results place classical algebraic Lissajous geometry in a setting where the decisive discriminating data are phase degeneracy and the arithmetic of the frequency pair $b\in\mathbb{C}^n$06 (Zhang, 2022).

4. Singular loci, nodal degeneracy, and discriminant-like criteria

A different strand of the literature studies discriminant-like phenomena through singularities, aliasing, and bad parameter loci. In multivariate interpolation, general Lissajous-Chebyshev nodes are described simultaneously as sampled points of suitable Lissajous curves and as points on a Chebyshev variety. For $b\in\mathbb{C}^n$07 and $b\in\mathbb{C}^n$08, the associated Chebyshev variety is

$b\in\mathbb{C}^n$09

The paper proves that this variety is the union of the images of an explicitly described family of Lissajous curves, and that the node set is obtained by equidistant sampling along those curves. Its singular points are characterized exactly by a coordinatewise criticality condition: $b\in\mathbb{C}^n$10 The same framework identifies a spectral aliasing relation on the Chebyshev index set and a notion of curve equivalence in which degenerate curves are precisely those satisfying $b\in\mathbb{C}^n$11. The paper does not define a named discriminant polynomial, but it explicitly treats singularities of the Chebyshev variety, aliasing classes, and curve degeneracy as the structural mechanisms organizing the theory (Dencker et al., 2017).

In knot theory, the relevant discriminant-like locus is the set of parameter values for which a Lissajous shadow becomes degenerate. A Fourier knot of type $b\in\mathbb{C}^n$12 is a closed embedded curve in $b\in\mathbb{C}^n$13 with finite Fourier sums in each coordinate, and a Lissajous knot is the special type $b\in\mathbb{C}^n$14. The paper proves that any knot in $b\in\mathbb{C}^n$15 is isotopic to a Fourier knot of type $b\in\mathbb{C}^n$16, obtained by deformation of a Lissajous knot. The planar shadow

$b\in\mathbb{C}^n$17

is analyzed under the assumptions that $b\in\mathbb{C}^n$18 are coprime and $b\in\mathbb{C}^n$19 is a small positive irrational number. A generic planar Lissajous curve then has

$b\in\mathbb{C}^n$20

nodes, parametrized by integer lattice points inside a triangle $b\in\mathbb{C}^n$21. The deformation

$b\in\mathbb{C}^n$22

produces perturbed node parameters whose $b\in\mathbb{C}^n$23-linear independence is controlled by a Wronskian criterion. The paper proves nonvanishing of that Wronskian under the arithmetic conditions

$b\in\mathbb{C}^n$24

with small $b\in\mathbb{C}^n$25 and $b\in\mathbb{C}^n$26 near $b\in\mathbb{C}^n$27. In this setting, the discriminant-like object is the bad arithmetic locus where degeneracy or rational dependence would obstruct the encoding of crossing data by a height function (Soret et al., 2015).

5. Quantum pattern selection and prime-frequency curve families

In quantum mechanics, Lissajous discrimination is realized through state selection rather than polynomial elimination. For the two-dimensional harmonic oscillator with commensurate frequencies

$b\in\mathbb{C}^n$28

the classical trajectories are

$b\in\mathbb{C}^n$29

The quantum construction starts from an ordinary two-mode coherent state and projects it onto a degenerate energy eigenspace. In the isotropic case $b\in\mathbb{C}^n$30, the resulting normalized projected state is

$b\in\mathbb{C}^n$31

In the anisotropic coprime case, the projected state depends on

$b\in\mathbb{C}^n$32

The probability density $b\in\mathbb{C}^n$33 localizes along the corresponding classical Lissajous figure. The distinguishing data are the frequency ratio $b\in\mathbb{C}^n$34, the relative phase $b\in\mathbb{C}^n$35, the amplitude ratio $b\in\mathbb{C}^n$36, and the current regime. Static states have $b\in\mathbb{C}^n$37 and strong interference fringes; vortex states have $b\in\mathbb{C}^n$38 but $b\in\mathbb{C}^n$39, with closed divergenceless current loops and weakened or absent fringes. Higher harmonics do not reduce to the same simple shape but become coherent superpositions of multiple fundamental quantum Lissajous states weighted by roots of unity (Russo, 2024).

The prime-frequency note gives a markedly different, explicitly exploratory use of discriminating features. It begins from the classical Lissajous/Bowditch curve

$b\in\mathbb{C}^n$40

and then introduces the finite Fourier-type prime-frequency curve

$b\in\mathbb{C}^n$41

where $b\in\mathbb{C}^n$42 are the first $b\in\mathbb{C}^n$43 prime numbers. The factor $b\in\mathbb{C}^n$44 is emphasized as necessary “to achieve a finite surface,” that is, to keep the sum bounded and visually manageable. The explicit examples are $b\in\mathbb{C}^n$45, $b\in\mathbb{C}^n$46, and $b\in\mathbb{C}^n$47, whose largest primes are $b\in\mathbb{C}^n$48, $b\in\mathbb{C}^n$49, and $b\in\mathbb{C}^n$50, respectively. The resulting curves exhibit a “slight left-right symmetry breaking,” and increasing $b\in\mathbb{C}^n$51 adds finer oscillatory structure. The paper also studies a separated subsequence construction using $b\in\mathbb{C}^n$52 and $b\in\mathbb{C}^n$53, reporting that this version converges much faster and that “it is not possible to see the differences between the two figures with naked eyes.” Attempts to enrich the structure by inserting logarithmic, square-root, and power-law functions into the arguments of the trigonometric terms were, in the authors’ words, “Unfortunately in vain.” The note does not define a formal discriminant; its distinguishing criteria are the choice of prime set, the growth of $b\in\mathbb{C}^n$54, the normalization by $b\in\mathbb{C}^n$55, the observed asymmetry, and the convergence behavior (Barna et al., 2020).

6. Nonlinear resonance, particular superintegrability, and the scope of the term

For nonlinear oscillators, “Lissajous discriminant” becomes a resonance criterion. The family

$b\in\mathbb{C}^n$56

generalizes harmonic Lissajous figures to separable polynomial potentials. The paper distinguishes three regimes. For the harmonic case $b\in\mathbb{C}^n$57,

$b\in\mathbb{C}^n$58

and the orbit closes precisely when

$b\in\mathbb{C}^n$59

The closed orbits admit an implicit Lissajous relation

$b\in\mathbb{C}^n$60

and the resonant system is maximally superintegrable because there exists an additional global integral $b\in\mathbb{C}^n$61 on all of phase space (Escobar-Ruiz et al., 23 Jun 2026).

For the quartic case $b\in\mathbb{C}^n$62, the oscillator is non-isochronous. The partial energies $b\in\mathbb{C}^n$63 remain global Liouville integrals, but the frequencies scale as

$b\in\mathbb{C}^n$64

The closed-orbit criterion is therefore

$b\in\mathbb{C}^n$65

so resonance depends on the initial data through the partial energies. On the equal-energy shell $b\in\mathbb{C}^n$66, one has $b\in\mathbb{C}^n$67 for the $b\in\mathbb{C}^n$68 resonances. The resulting orbit equation is algebraic and arises from Jacobi multiplication formulas: $b\in\mathbb{C}^n$69 The associated additional conserved quantities are not global integrals but particular integrals, satisfying

$b\in\mathbb{C}^n$70

They are conserved only on the resonant invariant manifold (Escobar-Ruiz et al., 23 Jun 2026).

For $b\in\mathbb{C}^n$71, the frequency ratio becomes

$b\in\mathbb{C}^n$72

and closed trajectories again require rational resonance. On the equal-energy shell $b\in\mathbb{C}^n$73, the $b\in\mathbb{C}^n$74 resonance condition reduces to

$b\in\mathbb{C}^n$75

The motion is described in terms of the generalized cosine $b\in\mathbb{C}^n$76, and orbit closure is encoded by a hyperelliptic phase constraint rather than a universal polynomial orbit equation: $b\in\mathbb{C}^n$77 The corresponding particular integrals are phase-locking functions such as

$b\in\mathbb{C}^n$78

In this nonlinear setting, the discriminating object is not a single polynomial discriminant but the resonance manifold together with its phase-locking condition (Escobar-Ruiz et al., 23 Jun 2026).

Across these contexts, the term “Lissajous discriminants” therefore has a layered meaning. In the strict algebraic-geometric sense it denotes the branch locus $b\in\mathbb{C}^n$79 of a projection associated with a Lissajous variety (Mascarin et al., 8 Sep 2025). In classical algebraic geometry it refers, more loosely, to the eliminant polynomials whose irreducibility is governed by $b\in\mathbb{C}^n$80 and phase degeneracy (Zhang, 2022). In interpolation, knot theory, prime-frequency constructions, quantum mechanics, and nonlinear dynamics, it denotes discriminant-like criteria that separate admissible from degenerate parameter regimes, distinguish one Lissajous family from another, or identify where topology, symmetry, localization, or periodicity changes (Dencker et al., 2017, Soret et al., 2015, Barna et al., 2020, Russo, 2024, Escobar-Ruiz et al., 23 Jun 2026).