  The field of fine-grained complexity aims at proving conditional lower bounds
on the time complexity of computational problems. One of the most popular
assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot
be solved in $2^{(1-\epsilon)n}$ time. In recent years, it has been proved that
known algorithms for many problems are optimal under SETH. Despite the wide
applicability of SETH, for many problems, there are no known SETH-based lower
bounds, so the quest for new reductions continues.
  Two barriers for proving SETH-based lower bounds are known. Carmosino et al.
(ITCS 2016) introduced the Nondeterministic Strong Exponential Time Hypothesis
(NSETH) stating that TAUT cannot be solved in time $2^{(1-\epsilon)n}$ even if
one allows nondeterminism. They used this hypothesis to show that some natural
fine-grained reductions would be difficult to obtain: proving that, say, 3-SUM
requires time $n^{1.5+\epsilon}$ under SETH, breaks NSETH and this, in turn,
implies strong circuit lower bounds. Recently, Belova et al. (SODA 2023)
introduced the so-called polynomial formulations to show that for many NP-hard
problems, proving any explicit exponential lower bound under SETH also implies
strong circuit lower bounds.
  We prove that for a range of problems from P, including $k$-SUM and triangle
detection, proving superlinear lower bounds under SETH is challenging as it
implies new circuit lower bounds. To this end, we show that these problems can
be solved in nearly linear time with oracle calls to evaluating a polynomial of
constant degree. Then, we introduce a strengthening of SETH stating that
solving SAT in time $2^{(1-\varepsilon)n}$ is difficult even if one has
constant degree polynomial evaluation oracle calls. This hypothesis is stronger
and less believable than SETH, but refuting it is still challenging: we show
that this implies circuit lower bounds.
