We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties. This framework provides clear graphical representations of algebraic structures -- such as polynomials, ideals, and varieties -- enabling intuitive yet rigorous diagrammatic reasoning. We showcase two practical viewpoints on GAG. First, we show that instances of counting constraint satisfaction problem (#CSP) are recast as rewrite problems of closed diagrams in GAG. This means that deciding rewritability in GAG is #P-hard, and GAG can be viewed as a complete and compositional rewrite system for networks of polynomial constraints. Second, we characterize the qudit ZH calculus, a diagrammatic language for quantum computation, as an extension of Graphical Algebraic Geometry. This establishes the correspondence that Graphical Algebraic Geometry is to the ZH calculus what Graphical Linear Algebra is to the ZX calculus. Using this construction, we show that computing amplitudes in qudit ZH requires only a constant number of queries to a GAG oracle.

翻译：我们引入图形代数几何（GAG），这是一族扩展了图形线性代数纲领的图解语言。我们在该族内构造了若干种语言，并证明了它们对于交换代数和仿射簇相应的（余）跨度语义的普适性与完备性。该框架为多项式、理想、簇等代数结构提供了清晰的图形表示，支持直观而严谨的图解推理。我们展示了GAG的两种实际视角。首先，我们证明计数约束满足问题（#CSP）的实例可被重新表述为GAG中闭合图的归约问题。这意味着判定GAG中的可归约性是#P-难的，并且GAG可被视为多项式约束网络的完备组合归约系统。其次，我们将量子计算的图解语言——qudit ZH演算——刻画为图形代数几何的扩展。这建立了以下对应关系：图形代数几何之于ZH演算，正如图形线性代数之于ZX演算。借助这一构造，我们证明计算qudit ZH中的振幅仅需对GAG预言机进行常数次查询。