We give new lower bounds for the fragments of the Ideal Proof System (IPS) introduced by Grochow and Pitassi [Joshua A. Grochow and Toniann Pitassi, 2018]. The Ideal Proof System is a central topic in algebraic proof complexity developed in the context of Nullstellensatz refutation [Paul Beame et al., 1994] and simulates Extended Frege efficiently. Our main results are as follows. - mult-IPS_{Lin'}: We prove nearly quadratic-size formula lower bound for multilinear refutation (over the Boolean hypercube) of a variant of the subset-sum axiom polynomial. Extending this, we obtain a nearly matching qualitative statement for a constant degree target polynomial. - IPS_{Lin'}: Over the fields of characteristic zero, we prove exponential-size sum-of-ROABPs lower bound for the refutation of a variant of the subset-sum axiom polynomial. The result also extends over the fields of positive characteristics when the target polynomial is suitably modified. The modification is inspired by the recent results [Tuomas Hakoniemi et al., 2024; Amik Raj Behera et al., 2025]. The mult-IPS_{Lin'} lower bound result is obtained by combining the quadratic-size formula lower bound technique of Kalorkoti [Kalorkoti, 1985] with some additional ideas. The proof technique of IPS_{Lin'} lower bound result is inspired by the recent lower bound result of Chatterjee, Kush, Saraf and Shpilka [Prerona Chatterjee et al., 2024].

We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree $d$ which requires homogeneous non-commutative circuit of size $Ω(d/\log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $Ω(nd)$, if $d\leq n$, or $Ω(nd \frac{\log n}{\log d})$, if $d\geq n$. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.

In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran & Perifel 2009, Malod 2011, Mahajan & Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotone VPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat’s definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not. Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrubeš & Yehudayoff (2021). In that context, we show that transparent polynomials of large sparsity are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.

The motivating question for this work is a long standing open problem, posed by Nisan (1991), regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question continues to remain open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by \Hrubes, Wigderson and Yehudayoff (2011)) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards separating formulas and ABPs in the non-commutative setting, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n-variate degree d abecedarian polynomial $f_{n,d}(x)$ such that 1. $f_{n, d}(x)$ can be computed by an abecedarian ABP of size O(nd); 2. any abecedarian formula computing $f_{n, \log n}(x)$ must have size that is super-polynomial in n. We also show that a super-polynomial lower bound against abecedarian formulas for $f_{\log n, n}(x)$ would separate the powers of formulas and ABPs in the non-commutative setting.

Parametric path problems arise independently in diverse domains, ranging from transportation to finance, where they are studied under various assumptions. We formulate a general path problem with relaxed assumptions, and describe how this formulation is applicable in these domains. We study the complexity of the general problem, and a variant of it where preprocessing is allowed. We show that when the parametric weights are linear functions, algorithms remain tractable even under our relaxed assumptions. Furthermore, we show that if the weights are allowed to be non-linear, the problem becomes NP-hard. We also study the multi-dimensional version of the problem where the weight functions are parameterized by multiple parameters. We show that even with two parameters, this problem is NP-hard.

We show that any Algebraic Branching Program (ABP) computing the polynomial ∑_{i=1}^n xⁿ_i has at least Ω(n²) vertices. This improves upon the lower bound of Ω(nlog n), which follows from the classical result of Baur and Strassen [Volker Strassen, 1973; Walter Baur and Volker Strassen, 1983], and extends the results of Kumar [Mrinal Kumar, 2019], which showed a quadratic lower bound for homogeneous ABPs computing the same polynomial. Our proof relies on a notion of depth reduction which is reminiscent of similar statements in the context of matrix rigidity, and shows that any small enough ABP computing the polynomial ∑_{i=1}^n xⁿ_i can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial ∑_{i=1}^n xⁿ_i + ε(𝐱), for a structured "error polynomial" ε(𝐱). To complete the proof, we then observe that the lower bound in [Mrinal Kumar, 2019] is robust enough and continues to hold for all polynomials ∑_{i=1}^n xⁿ_i + ε(𝐱), where ε(𝐱) has the appropriate structure.

For every constant , we show that there is a family {PN, c} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following properties: For every family {fn} of polynomials in VP, where fn is an n variate polynomial of degree at most ncwith bounded integer coefficients and for N=nc+nn, PN, c vanishes on the coefficient vector of fn. There exists a family {hn} of polynomials where hn is an n variate polynomial of degree at most ncwith bounded integer coefficients such that for N=nc+nn, PN, c does not vanish on the coefficient vector of hn. In other words, there are efficiently computable equations for polynomials in VP that have small integer coefficients. In fact, we also prove an analogous statement for the seemingly larger class VNP. Thus, in this setting of polynomials with small integer coefficients, this provides evidence against a natural proof like barrier for proving algebraic circuit lower bounds, a framework for which was proposed in the works of Forbes, Shpilka and Volk [1], and Grochow, Kumar, Saks and Saraf [2]. Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes and also extend to finite fields of small size. Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes and also extend to finite fields of small size.

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [Malte Beecken et al., 2013] and exploited by them and Agrawal et al. [Manindra Agrawal et al., 2016] to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields were unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey, Saxena and Sinhababu [Anurag Pandey et al., 2018], we construct explicit faithful maps for some natural classes of polynomials in fields of positive characteristic, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken, Mittmann and Saxena [Malte Beecken et al., 2013] and Agrawal, Saha, Saptharishi, Saxena [Manindra Agrawal et al., 2016] in the positive characteristic setting.