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Revision #1 to TR18-135 | 12th June 2019 09:05
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#### Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

**Abstract:**
We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2016). We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms and injective directed homomorphisms and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties:

* The polynomial families complete for VF, VBP, and VP are model independent, i.e. they do not use a particular instance of a formula, ABP or circuit for characterising VF, VBP, or VP, respectively.

* All the polynomial families are hard under $p$-projections.

**Changes to previous version:**
*We have added one more polynomial family complete for VP.

*This family is essentially a generalisation of a polynomial family in the previous version.

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TR18-135 | 31st July 2018 20:11
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#### Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

**Abstract:**
We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2016). We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms and injective directed homomorphisms and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties:

* The polynomial families complete for VF, VBP, and VP are model independent, i.e. they do not use a particular instance of a formula, ABP or circuit for characterising VF, VBP, or VP, respectively.

* All the polynomial families are hard under $p$-projections.