Singer trace planes and cyclic difference sets

Abstract

The trace-zero points in the projective plane over a prime field form a cyclic Singer difference set of size p+1. Every nonzero cyclic difference occurs exactly once. Multiplication of indices by two permutes the odd-length cycle, so its inverse image has the same cardinality and difference multiplicities.

Theorem 1.1 (Multiplication has no proper invariant subspace).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall a:\mathrm{GaloisField}\left(p,3\right),\forall W:\mathrm{Submodule}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right)\right),(\neg (a\in \mathrm{Set}.\mathrm{range}\left(\mathrm{algebraMap}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right)\right)\right)))\Rightarrow ((\forall x:\mathrm{GaloisField}\left(p,3\right),(x\in W)\Rightarrow (a\cdot x\in W))\Rightarrow ((W=\mathrm{Bot}.\mathrm{bot})\vee (W=\mathrm{Top}.\mathrm{top})))$$

Proof. Machine-checked in Lean as D5/S3/Geometry/FiniteGeometry/SingerTracePlane.no_proper_invariant_subspace (✓ std3). ∎

Citation. James Singer (1938). A theorem in finite projective geometry and some applications to number theory. DOI: 10.1090/s0002-9947-1938-1501951-4. URL: https://doi.org/10.1090/s0002-9947-1938-1501951-4.

Commentary.

For an element outside the base field, a subspace preserved by multiplication is either zero or the whole cubic extension. The scalars preserving the subspace form a subalgebra; prime extension degree forces a non-base subalgebra to be the whole field.

Theorem 1.2 (Distinct trace planes meet in a line).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall a:\mathrm{GaloisField}\left(p,3\right),(\neg (a\in \mathrm{Set}.\mathrm{range}\left(\mathrm{algebraMap}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right)\right)\right)))\Rightarrow (\mathrm{Module}.\mathrm{finrank}\left(\mathrm{ZMod}\left(p\right),\mathrm{Subtype}\left((x:\mathrm{GaloisField}\left(p,3\right))\mapsto x\in \mathrm{Min}.\min \left((\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right)\right)).\ker ,(\mathrm{LinearMap}.\mathrm{comp}\left(\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right)\right),\mathrm{LinearMap}.\mathrm{mulLeft}\left(\mathrm{ZMod}\left(p\right),a\right)\right)).\ker \right)\right)\right)=1)$$

Proof. Machine-checked in Lean as D5/S3/Geometry/FiniteGeometry/SingerTracePlane.trace_plane_intersection (✓ std3). ∎

Citation. James Singer (1938). A theorem in finite projective geometry and some applications to number theory. DOI: 10.1090/s0002-9947-1938-1501951-4. URL: https://doi.org/10.1090/s0002-9947-1938-1501951-4.

Commentary.

Trace is surjective, so its kernel is a plane. The kernel of x mapped to Tr(a*x) is another plane for nonzero a. If a is outside the base field, equality of these planes would make the trace kernel invariant under multiplication by a. Their sum has dimension three, and their intersection dimension one.

Theorem 1.3 (The Singer difference-set parameters).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),(\mathrm{orderOf}\left(\alpha \right)=p^3-1)\Rightarrow ((\mathrm{Finset}.\mathrm{card}\left(\mathrm{Finset}.\mathrm{filter}\left((i:\mathrm{Fin}\left(p^2+p+1\right))\mapsto \mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{\mathrm{val}\left(i\right)}\right)=0,\mathrm{Finset}.\mathrm{univ}\left(\mathrm{Fin}\left(p^2+p+1\right)\right)\right)\right)=p+1)\wedge ((\mathrm{Finset}.\mathrm{card}\left(\mathrm{Finset}.\mathrm{filter}\left((i:\mathrm{Fin}\left(p^2+p+1\right))\mapsto \mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{2\cdot \mathrm{val}\left(i\right)}\right)=0,\mathrm{Finset}.\mathrm{univ}\left(\mathrm{Fin}\left(p^2+p+1\right)\right)\right)\right)=p+1)\wedge ((\forall r:\mathrm{Fin}\left(p^2+p+1\right),(r\ne 0)\Rightarrow (\mathrm{Finset}.\mathrm{card}\left(\mathrm{Finset}.\mathrm{filter}\left((i:\mathrm{Fin}\left(p^2+p+1\right))\mapsto (\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{\mathrm{val}\left(i\right)}\right)=0)\wedge (\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{\mathrm{val}\left(i+r\right)}\right)=0),\mathrm{Finset}.\mathrm{univ}\left(\mathrm{Fin}\left(p^2+p+1\right)\right)\right)\right)=1))\wedge ((\forall i:\mathrm{Fin}\left(p^2+p+1\right),(\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{2\cdot \mathrm{val}\left(i\right)}\right)=0)\iff (\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{\mathrm{val}\left(2\cdot i\right)}\right)=0))\wedge (\forall r:\mathrm{Fin}\left(p^2+p+1\right),(r\ne 0)\Rightarrow (\mathrm{Finset}.\mathrm{card}\left(\mathrm{Finset}.\mathrm{filter}\left((i:\mathrm{Fin}\left(p^2+p+1\right))\mapsto (\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{2\cdot \mathrm{val}\left(i\right)}\right)=0)\wedge (\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{2\cdot \mathrm{val}\left(i+r\right)}\right)=0),\mathrm{Finset}.\mathrm{univ}\left(\mathrm{Fin}\left(p^2+p+1\right)\right)\right)\right)=1))))))$$

Proof. Machine-checked in Lean as D5/S3/Geometry/FiniteGeometry/SingerTracePlane.result (✓ std3). ∎

Citation. James Singer (1938). A theorem in finite projective geometry and some applications to number theory. DOI: 10.1090/s0002-9947-1938-1501951-4. URL: https://doi.org/10.1090/s0002-9947-1938-1501951-4.

Commentary.

Singer (1938), pages 377-385: trace-zero projective points form the cyclic (p^2+p+1,p+1,1) difference set. Primitive powers enumerate projective points. A nonzero cyclic shift corresponds to a scalar outside the base field; the unique projective point of the trace-plane intersection counts its difference multiplicity. The last two clauses identify the trace-square support with the inverse image under doubling and give the same multiplicities. Subtraction and addition in Fin(p^2+p+1) are cyclic operations; val is the canonical natural representative.

References

• Truth anchor: D5/S3/Geometry/FiniteGeometry/SingerTracePlane.no_proper_invariant_subspace
• Truth anchor: D5/S3/Geometry/FiniteGeometry/SingerTracePlane.result
• Truth anchor: D5/S3/Geometry/FiniteGeometry/SingerTracePlane.trace_plane_intersection